# On Weak Annihilators and Nilpotent Associated Primes of Skew PBW Extensions

Sebastián Higuera  Universidad Nacional de Colombia - Sede Bogotá Campus
Universitario<EMAIL_ADDRESS>and  Armando Reyes  Universidad Nacional
de Colombia - Sede Bogotá Campus Universitario<EMAIL_ADDRESS>Dedicated
to the memory of Professor V. A. Artamonov

###### Abstract.

We investigate the notions of weak annihilator and nilpotent associated prime
defined by Ouyang and Birkenmeier [82] in the setting of noncommutative rings
having PBW bases. We extend several results formulated in the literature
concerning annihilators and associated primes of commutative rings and skew
polynomial rings to a more general setting of algebras not considered before.
We exemplify our results with families of algebras appearing in the theory of
enveloping algebras, differential operators on noncommutative spaces,
noncommutative algebraic geometry, and theoretical physics. Finally, we
present some ideas for future research.

###### Key words and phrases:

Weak annihilator, nilpotent associated prime, nilpotent good polyomial, skew
PBW extension

###### 2020 Mathematics Subject Classification:

16D25, 16N40, 16S30, 16S32, 16S36, 16S38

This work was supported by Faculty of Science, Universidad Nacional de
Colombia - Sede Bogotá, Bogotá, D. C., Colombia [grant number 52464].

##  1\. Introduction

Throughout the paper, every ring is associative (not necessarily commutative)
with identity unless otherwise stated. For a ring $R$, $P(R)$, $N(R)$, and
$N^{*}(R)$ denote the prime radical of $R$, the set of all nilpotent elements
of $R$, and the upper radical $N^{*}(R)$ of $R$ (i.e., the sum of all its nil
ideals of $R$), respectively. It is well-known that $N^{*}(R)\subseteq N(R)$,
and if the equality $N^{*}(R)=N(R)$ holds, Marks [75] called $R$ an NI ring.
Due to Birkenmeier et al. [21], $R$ is called 2-primal if $P(R)=N(R)$ (Sun
[105] used the term _weakly symmetric_ for these rings). Notice that every
reduced ring (a ring which has no non-zero nilpotent elements) is 2-primal.
The importance of 2-primal rings is that they can be considered as a
generalization of commutative rings and reduced rings. For more details about
2-primal rings, see Birkenmeier et al. [22] and Marks [76].

For a subset $B$ of a ring $R$, the sets $r_{R}(B)=\\{r\in R\mid Br=0\\}$ and
$l_{R}(B)=\\{r\in R\mid rB=0\\}$ represent the right and left annihilator of
$B$ in $R$, respectively. Properties of annihilators of subsets in rings have
been investigated by several authors [12, 30, 31, 62, 110]. For example,
Kaplansky [62] introduced the Baer rings as those rings for which the right
(left) annihilator of every nonempty subset of the ring is generated by an
idempotent element. This concept has its roots in functional analysis, having
close links to $C^{*}$-algebras and von Neumann algebras (e.g., Berberian
[17]). Closely related to Baer rings are the quasi-Baer rings. A ring $R$ is
said to be quasi-Baer if the left annihilator of an ideal of $R$ is generated
by an idempotent. Now, $R$ is called right (left) p.p. (principally
projective) or left Rickart ring if the right (left) annihilator of each
element of $R$ is generated by an idempotent (equivalently, any principal left
ideal of the ring is projective). Birkenmeier [20] defined a ring $R$ to be
right (left) principally quasi-Baer (or simply right (left) p.q.-Baer) ring if
the right annihilator of each principal right (left) ideal of $R$ is generated
(as a right ideal) by an idempotent. Families of rings satisfying Baer, quasi-
Baer, p.p. and p.q.-Baer conditions have been presented in the literature. For
instance, Armendariz [7] established that if $R$ is a reduced ring, then the
commutative polynomial ring $R[x]$ is a Baer ring if and only if $R$ is a Baer
ring [7, Theorem B]. Birkenmeier et al. [20] showed that the quasi-Baer
condition is preserved by many polynomial extensions and proved that a ring
$R$ is right p.q.-Baer if and only if $R[x]$ is right p.q.-Baer. For more
details about these ring-theoretical properties, commutative and
noncommutative examples, see [20, 44, 45, 50, 94], and references therein, and
the excellent treatment by Birkenmeier et al. [22].

As a generalization of right and left annihilators, Ouyang and Birkenmeier
[82] introduced the notion of weak annihilator of a subset in a ring, and
investigated its properties over Ore extensions (also known as skew polynomial
rings) $R[x;\sigma,\delta]$ defined by Ore [86]. More exactly, for $R$ a ring
and $B$ a subset of $R$, they defined $N_{R}(B)=\\{r\in R\mid br\in N(R),\
{\rm for\ all}\ b\in B\\}$, which is called the weak annihilator of $B$ in $R$
[82, Definition 2.1]. The theory developed by them makes use of the notion of
compatible ring in the sense of Annin [3] (see also Hashemi and Moussavi
[45]). Briefly, if $R$ is a ring, $\sigma$ is an endomorphism of $R$, and
$\delta$ is a $\sigma$-derivation of $R$, then (i) $R$ is said to be
$\sigma$-compatible if for each $a,b\in R$, $ab=0$ if and only if
$a\sigma(b)=0$ (necessarily, the endomorphism $\sigma$ is injective). (ii) $R$
is called $\delta$-compatible if for each $a,b\in R$, $ab=0$ implies
$a\delta(b)=0$. (iii) If $R$ is both $\sigma$-compatible and
$\delta$-compatible, then $R$ is called ($\sigma,\delta$)-compatible.

For $R$ a $(\sigma,\delta)$-compatible 2-primal ring, Ouyang and Birkenmeier
[82] proved the following: (i) if for each subset $B\nsubseteq N(R)$,
$N_{R}(B)$ is generated as an ideal by a nilpotent element, then for each
subset $U\nsubseteq N(R[x;\sigma,\delta])$, $N_{R[x;\sigma,\delta]}(U)$ is
generated as an ideal by a nilpotent element [82, Theorem 2.1]. (ii) The
following two statements are equivalent: (1) for each subset $B\nsubseteq
N(R)$, $N_{R}(B)$ is generated as an ideal by a nilpotent element. (2) For
each subset $U\nsubseteq N(R[x;\sigma])$, $N_{R[x;\sigma]}(U)$ is generated as
an ideal by a nilpotent element [82, Theorem 2.2]. (iii) If for each principal
right ideal $pR\nsubseteq N(R)$, $N_{R}(pR)$ is generated as an ideal by a
nilpotent element, then for each principal right ideal
$f(x)R[x;\sigma,\delta]\nsubseteq N(R[x;\sigma,\delta])$,
$N_{R[x;\sigma,\delta]}(f(x)R[x;\sigma,\delta])$ is generated as an ideal by a
nilpotent element [82, Theorem 2.3]. (iv) The following statements are
equivalent: (1) For each principal right ideal $pR\nsubseteq N(R)$,
$N_{R}(pR)$ is generated as an ideal by a nilpotent element. (2) For each
principal right ideal $f(x)R[x;\sigma]\nsubseteq N(R)$,
$N_{R[x;\sigma]}(f(x)R[x;\sigma])$ is generated as an ideal by a nilpotent
element [82, Theorem 2.4]. (v) If for each $p\notin N(R)$, $N_{R}(p)$ is
generated as an ideal by a nilpotent element, then for each $f(x)\notin
N(R[x;\sigma,\delta])$, $N_{R[x;\sigma,\delta]}(f(x))$ is generated as an
ideal by a nilpotent element [82, Theorem 2.5]. (vi) The following statements
are equivalent: (1) for each $p\notin N_{R}(p)$, $N_{R}(p)$ is generated as an
ideal by a nilpotent element. (2) For each skew polynomial $f(x)\notin
N(R[x;\sigma])$, $N_{R[x;\sigma]}(f(x))$ is generated as an ideal by a
nilpotent element.

With the aim of investigating if the previous results obtained by Ouyang and
Birkenmeier hold in noncommutative polynomial extensions more general than Ore
extensions, in this paper we consider the class of skew Poincaré-Birkhoff-Witt
extensions introduced by Gallego and Lezama [35] because they include families
of rings appearing in noncommutative algebra and noncommutative geometry (see
Section 2 for a description of the generality of these objects with respect to
another families of noncommutative rings). Ring-theoretical, homological and
geometrical properties of these objects have been investigated by some people
(e.g. [8, 33, 42, 48, 69, 72, 104, 106]).

The paper is organized as follows. In Section 2, we recall some definitions
and results about skew PBW extensions and compatible rings for finite families
of endomorphisms and derivations of rings. Section 3 contains the original
results of the article concerning weak annihilator ideals of skew PBW
extensions (Theorems 3.4, 3.5, 3.8, 3.10, 3.13, 3.15, and 3.18). Next, in
Section 4, we characterize the nilpotent associated prime ideals of skew PBW
extensions (Theorems 4.3 and 4.4). As expected, our results generalize those
above corresponding skew polynomial extensions presented by Ouyang et al. [82,
84]. It is worth mentioning that this work is a sequel of the study of ideals
of skew PBW extensions that has been realized by different authors (e.g. [42,
70, 73, 80, 92, 96, 98]). In this way, the results formulated in this paper
about associated primes extend or contribute to those presented by Annin [1,
2, 3], Bhat [18, 19], Brewer and Heinzer [23], Faith [31], Leroy and Matczuk
[68], and Niño et al. [79]. Finally, Section 5 illustrate the results
established in Sections 3 and 4 with several noncommutative algebras that
cannot be expressed as Ore extensions.

Throughout the paper, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, and
$\mathbb{C}$ denote the classical numerical systems. We assume the set of
natural numbers including zero. The symbol $\Bbbk$ denotes a field and
$\Bbbk^{*}:=\Bbbk\ \backslash\ \\{0\\}$.

##  2\. Skew Poincaré-Birkhoff-Witt extensions

Skew PBW extensions were defined by Gallego and Lezama [35] with the aim of
generalizing Poincaré-Birkhoff-Witt extensions introduced by Bell and Goodearl
[13] and Ore extensions of injective type defined by Ore in [86]. Over the
years, several authors have shown that skew PBW extensions also generalize
families of noncommutative algebras such as 3-dimensional skew polynomial
algebras introduced by Bell and Smith [14], diffusion algebras defined by
Isaev et al. [51], ambiskew polynomial rings introduced by Jordan in several
papers [52, 53, 54, 55, 56, 57], solvable polynomial rings introduced by
Kandri-Rody and Weispfenning [61], almost normalizing extensions defined by
McConnell and Robson [77], skew bi-quadratic algebras recently introduced by
Bavula [11], and others (see [33] for more details).

The importance of skew PBW extensions is that they do not assume that the
coefficients commute with the variables, and the coefficients do not
necessarily belong to fields (see Definition 2.1). As a matter of fact, skew
PBW extensions contain well-known groups of algebras such as some types of
$G$-algebras in the sense of Apel [6], Auslander-Gorenstein rings, some
Calabi-Yau and skew Calabi-Yau algebras, some Artin-Schelter regular algebras,
some Koszul algebras, quantum polynomials, some quantum universal enveloping
algebras, families of differential operator rings, and many other algebras of
great interest in noncommutative algebraic geometry and noncommutative
differential geometry. For more details about relations between skew PBW
extensions and other noncommutative algebras having PBW bases, see [24, 33,
37, 72, 102], and references therein.

######  Definition 2.1 ([35, Definition 1]).

Let $R$ and $A$ be rings. We say that $A$ is a skew PBW extension (also known
as $\sigma$-PBW extension) over $R$, which is denoted by $A:=\sigma(R)\langle
x_{1},\dots,x_{n}\rangle$, if the following conditions hold:

  1. (i)

$R$ is a subring of $A$ sharing the same multiplicative identity element.

  2. (ii)

there exist elements $x_{1},\dots,x_{n}\in A$ such that $A$ is a left free
$R$-module with basis given by ${\rm
Mon}(A):=\\{x^{\alpha}=x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}\mid\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}\\}$,
where $x_{1}^{0}\dotsb x_{n}^{0}:=1$.

  3. (iii)

For each $1\leq i\leq n$ and any $r\in R\ \backslash\ \\{0\\}$, there exists
an element $c_{i,r}\in R\ \backslash\ \\{0\\}$ such that
$x_{i}r-c_{i,r}x_{i}\in R$.

  4. (iv)

For any elements $1\leq i,j\leq n$, there exists $d_{i,j}\in R\ \backslash\
\\{0\\}$ such that $x_{j}x_{i}-d_{i,j}x_{i}x_{j}\in R+Rx_{1}+\cdots+Rx_{n}$
(i.e., there exist elements $r_{0}^{(i,j)},r_{1}^{(i,j)},\dotsc,r_{n}^{(i,j)}$
of $R$ with
$x_{j}x_{i}-d_{i,j}x_{i}x_{j}=r_{0}^{(i,j)}+\sum_{k=1}^{n}r_{k}^{(i,j)}x_{k}$).

From Definition 2 it follows that every non-zero element $f\in A$ can be
uniquely expressed as $f=a_{0}X_{0}+a_{1}X_{1}+\cdots+a_{m}X_{m}$, with
$a_{i}\in R$ and $X_{i}\in\text{Mon}(A)$, for $0\leq i\leq m$ ($X_{0}:=1$)
[35, Remark 2].

Thinking about Ore extensions [86], if $A=\sigma(R)\langle
x_{1},\dots,x_{n}\rangle$ is a skew PBW extension over $R$, then for every
$1\leq i\leq n$ there exist an injective endomorphism $\sigma_{i}:R\rightarrow
R$ and a $\sigma_{i}$-derivation $\delta_{i}:R\rightarrow R$ such that
$x_{i}r=\sigma_{i}(r)x_{i}+\delta_{i}(r)$, for each $r\in R$ [35, Proposition
3]. When necessary, we will write $\Sigma:=\\{\sigma_{1},\dotsc,\sigma_{n}\\}$
and $\Delta:=\\{\delta_{1},\dotsc,\delta_{n}\\}$.

######  Definition 2.2.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension over
$R$.

  1. (i)

([35, Definition 4]) $A$ is called quasi-commutative if the conditions (iii)
and (iv) in Definition 2.1 are replaced by the following: (iii’) for each
$1\leq i\leq n$ and all $r\in R\ \backslash\ \\{0\\}$, there exists
$c_{i,r}\in R\ \backslash\ \\{0\\}$ such that $x_{i}r=c_{i,r}x_{i}$. (iv’) for
any $1\leq i,j\leq n$, there exists $d_{i,j}\in R\ \backslash\ \\{0\\}$ such
that $x_{j}x_{i}=d_{i,j}x_{i}x_{j}$.

  2. (ii)

([35, Definition 4]) $A$ is called bijective if $\sigma_{i}$ is bijective for
each $1\leq i\leq n$, and $d_{i,j}$ is invertible, for any $1\leq i,j\leq n$.

  3. (iii)

([70, Definition 2.3]) $A$ is called of endomorphism type if $\delta_{i}=0$,
for every $i$. In addition, if every $\sigma_{i}$ is bijective, then $A$ is
said to be a skew PBW extension of automorphism type.

Some relations between skew polynomial rings [86], PBW extensions [13] and
skew PBW extensions are presented below.

######  Remark 2.3.

  * (i)

([71, Example 5(3)]) If
$R[x_{1};\sigma_{1},\delta_{1}]\dotsb[x_{n};\sigma_{n},\delta_{n}]$ is an
iterated Ore extension such that $\sigma_{i}$ is injective for $1\leq i\leq
n$; $\sigma_{i}(r)$, $\delta_{i}(r)\in R$, for $r\in R$ and $1\leq i\leq n$;
$\sigma_{j}(x_{i})=c_{i}x_{i}+d_{i}$, for $i<j$, $c_{i},d_{i}\in R$, where $c$
has a left inverse; $\delta_{j}(x_{i})\in R+Rx_{1}+\dotsb+Rx_{n}$, for $i<j$,
then
$R[x_{1};\sigma_{1},\delta_{1}]\dotsb[x_{n};\sigma_{n},\delta_{n}]\cong\sigma(R)\langle
x_{1},\dotsc,x_{n}\rangle$.

  * (ii)

Skew polynomial rings are not included in PBW extensions. For instance, the
quantum plane defined as the quotient $\Bbbk\langle x,y\rangle/\langle xy-
qyx\mid q\in\Bbbk^{*}\rangle$ is an Ore extension of injective type given by
$\Bbbk[y][x;\sigma]$, where $\sigma(y)=qy$, but it is clear that this cannot
be expressed as a PBW extension over $\Bbbk$ or $\Bbbk[y]$.

  * (iii)

Skew PBW extensions are not contained in skew polynomial rings. For example,
the universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra
$\mathfrak{g}$ is a PBW extension ([13, Section 5]), and hence a skew PBW
extension, but in general it cannot be expressed as an Ore extension.

  * (iii)

([71, Theorem 2.3]) If $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ is a
quasi-commutative skew PBW extension over a ring $R$, then $A$ is isomorphic
to an iterated Ore extension of endomorphism type (i.e., that is,
$R[x;\sigma]$ with $\sigma$ surjective).

  * (iv)

([48, Remark 3.4]) Skew PBW extensions of endomorphism type are more general
than iterated Ore extensions of endomorphism type. Let us illustrate the
situation with two and three indeterminates.

For the iterated Ore extension of endomorphism type
$R[x;\sigma_{x}][y;\sigma_{y}]$, if $r\in R$ then we have the following
relations: $xr=\sigma_{x}(r)x$, $yr=\sigma_{y}(r)y$, and $yx=\sigma_{y}(x)y$.
Now, if we have $\sigma(R)\langle x,y\rangle$ a skew PBW extension of
endomorphism type over $R$, then for any $r\in R$, Definition 2.1 establishes
that $xr=\sigma_{1}(r)x$, $yr=\sigma_{2}(r)y$, and
$yx=d_{1,2}xy+r_{0}+r_{1}x+r_{2}y$, for some elements $d_{1,2},r_{0},r_{1}$
and $r_{2}$ belong to $R$. From these relations it is clear which one of them
is more general.

If we have the iterated Ore extension
$R[x;\sigma_{x}][y;\sigma_{y}][z;\sigma_{z}]$, then for any $r\in R$,
$xr=\sigma_{x}(r)x$, $yr=\sigma_{y}(r)y$, $zr=\sigma_{z}(r)z$,
$yx=\sigma_{y}(x)y$, $zx=\sigma_{z}(x)z$, $zy=\sigma_{z}(y)z$. For the skew
PBW extension of endomorphism type $\sigma(R)\langle x,y,z\rangle$,
$xr=\sigma_{1}(r)x$, $yr=\sigma_{2}(r)y$, $zr=\sigma_{3}(r)z$,
$yx=d_{1,2}xy+r_{0}+r_{1}x+r_{2}y+r_{3}z$,
$zx=d_{1,3}xz+r_{0}^{\prime}+r_{1}^{\prime}x+r_{2}^{\prime}y+r_{3}^{\prime}z$,
and
$zy=d_{2,3}yz+r_{0}^{\prime\prime}+r_{1}^{\prime\prime}x+r_{2}^{\prime\prime}y+r_{3}^{\prime\prime}z$,
for some elements
$d_{1,2},d_{1,3},d_{2,3},r_{0},r_{0}^{\prime},r_{0}^{\prime\prime},r_{1},r_{1}^{\prime},r_{1}^{\prime\prime},r_{2},r_{2}^{\prime},r_{2}^{\prime\prime},r_{3}$,
$r_{3}^{\prime},r_{3}^{\prime\prime}$ of $R$. As the number of indeterminates
increases, the differences between both algebraic structures are more
remarkable.

Following [35, Section 3], if $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is
a skew PBW extension over $R$, then we consider the following notation:

  1. (i)

For the families $\Sigma$ and $\Delta$ in Definition 2.1, from now on, if
$\alpha=(\alpha_{1},\dots,\alpha_{n})$ belongs to $\mathbb{N}^{n}$, we will
write
$\sigma^{\alpha}:=\sigma_{1}^{\alpha_{1}}\circ\dotsb\circ\sigma_{n}^{\alpha_{n}}$,
$\sigma^{-\alpha}:=\sigma_{n}^{\alpha_{n}}\circ\dotsb\circ\sigma_{1}^{\alpha_{1}}$
and
$\delta^{\alpha}=\delta_{1}^{\alpha_{1}}\circ\dotsb\circ\delta_{n}^{\alpha_{n}}$,
where $\circ$ denotes the classical composition of functions, and
$|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$. If
$\beta=(\beta_{1},\dots,\beta_{n})\in\mathbb{N}^{n}$, then
$\alpha+\beta:=(\alpha_{1}+\beta_{1},\dots,\alpha_{n}+\beta_{n})$.

  2. (ii)

Let $\succeq$ be a total order defined on ${\rm Mon}(A)$. If
$x^{\alpha}\succeq x^{\beta}$ but $x^{\alpha}\neq x^{\beta}$, we will write
$x^{\alpha}\succ x^{\beta}$. As we saw above, if $f$ is a non-zero element of
$A$, then $f$ can be expressed uniquely as
$f=a_{0}X_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}$, with $a_{i}\in R$,
$X_{m}\succ\dotsb\succ X_{1}$, and $X_{0}:=1$. Eventually, we use expressions
as $f=a_{0}Y_{0}+a_{1}Y_{1}+\dotsb+a_{m}Y_{m}$, with $a_{i}\in R$, and
$Y_{m}\succ\dotsb\succ Y_{1}$. We define ${\rm lm}(f):=X_{m}$, the leading
monomial of $f$; ${\rm lc}(f):=a_{m}$, the leading coefficient of $f$; ${\rm
lt}(f):=a_{m}X_{m}$, the leading term of $f$; ${\rm exp}(f):={\rm
exp}(X_{m})$, the order of $f$. Notice that $\deg(f):={\rm
max}\\{\deg(X_{i})\\}_{i=1}^{m}$. If $f=0$, then ${\rm lm}(0):=0$, ${\rm
lc}(0):=0$, ${\rm lt}(0):=0$. Finally, we also consider $X\succ 0$ for any
$X\in{\rm Mon}(A)$, and hence we extend $\succeq$ to ${\rm
Mon}(A)\cup\\{0\\}$.

From [35, Definition 11], if $\succeq$ is a total order on ${\rm Mon}(A)$,
then we say that $\succeq$ is a monomial order on ${\rm Mon}(A)$ if the
following conditions hold:

     * •

For every $x^{\beta},x^{\alpha},x^{\gamma},x^{\lambda}\in{\rm Mon}(A)$, the
relation $x^{\beta}\succeq x^{\alpha}$ implies ${\rm
lm}(x^{\gamma}x^{\beta}x^{\lambda})\succeq{\rm
lm}(x^{\gamma}x^{\alpha}x^{\lambda})$ (i.e., the total order is compatible
with multiplication).

     * •

$x^{\alpha}\succeq 1$, for every $x^{\alpha}\in{\rm Mon}(A)$.

     * •

$\succeq$ is degree compatible, i.e., $|\beta|\succeq|\alpha|\Rightarrow
x^{\beta}\succeq x^{\alpha}$.

Monomial orders are also called admissible orders. The condition (iii) of the
previous definition is needed in the proof of the fact that every monomial
order on ${\rm Mon}(A)$ is a well order, that is, there are not infinite
decreasing chains in ${\rm Mon}(A)$ [35, Proposition 12].

  3. (iii)

For a skew PBW extension $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ over a
ring $R$ and $B$ a subset of $R$, $BA$ denotes the set
$\bigl{\\{}f=\sum_{i=0}^{m}b_{i}X_{i}\mid b_{i}\in B,\ {\rm for\ all}\
i\bigr{\\}}$.

Propositions 2.4 and 2.5 are very useful in the calculations presented in
Sections 3 and 4.

######  Proposition 2.4 ([35, Theorem 7]).

If $A$ is a polynomial ring with coefficients in $R$ with respect to the set
of indeterminates $\\{x_{1},\dots,x_{n}\\}$, then $A=\sigma(R)\langle
x_{1},\dots,x_{n}\rangle$ is a skew PBW extension over $R$ if and only if the
following conditions hold:

  1. (1)

for each $x^{\alpha}\in{\rm Mon}(A)$ and every $0\neq r\in R$, there exist
unique elements $r_{\alpha}:=\sigma^{\alpha}(r)\in R\ \backslash\ \\{0\\}$,
$p_{\alpha,r}\in A$, such that
$x^{\alpha}r=r_{\alpha}x^{\alpha}+p_{\alpha,r}$, where $p_{\alpha,r}=0$, or
$\deg(p_{\alpha,r})<|\alpha|$ if $p_{\alpha,r}\neq 0$. If $r$ is left
invertible, so is $r_{\alpha}$.

  2. (2)

For each $x^{\alpha},x^{\beta}\in{\rm Mon}(A)$, there exist unique elements
$d_{\alpha,\beta}\in R$ and $p_{\alpha,\beta}\in A$ such that
$x^{\alpha}x^{\beta}=d_{\alpha,\beta}x^{\alpha+\beta}+p_{\alpha,\beta}$, where
$d_{\alpha,\beta}$ is left invertible, $p_{\alpha,\beta}=0$, or
$\deg(p_{\alpha,\beta})<|\alpha+\beta|$ if $p_{\alpha,\beta}\neq 0$.

######  Proposition 2.5 ([92, Proposition 2.7 and Remark 2.8]).

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a skew PBW extension
over $R$. If $\alpha=(\alpha_{1},\dotsc,\alpha_{n})\in\mathbb{N}^{n}$ and $r$
is an element of $R$, then

$\displaystyle x^{\alpha}r=$ $\displaystyle\
x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\dotsb
x_{n-1}^{\alpha_{n-1}}x_{n}^{\alpha_{n}}r=x_{1}^{\alpha_{1}}\dotsb
x_{n-1}^{\alpha_{n-1}}\biggl{(}\sum_{j=1}^{\alpha_{n}}x_{n}^{\alpha_{n}-j}\delta_{n}(\sigma_{n}^{j-1}(r))x_{n}^{j-1}\biggr{)}$
$\displaystyle+$ $\displaystyle\ x_{1}^{\alpha_{1}}\dotsb
x_{n-2}^{\alpha_{n-2}}\biggl{(}\sum_{j=1}^{\alpha_{n-1}}x_{n-1}^{\alpha_{n-1}-j}\delta_{n-1}(\sigma_{n-1}^{j-1}(\sigma_{n}^{\alpha_{n}}(r)))x_{n-1}^{j-1}\biggr{)}x_{n}^{\alpha_{n}}$
$\displaystyle+$ $\displaystyle\ x_{1}^{\alpha_{1}}\dotsb
x_{n-3}^{\alpha_{n-3}}\biggl{(}\sum_{j=1}^{\alpha_{n-2}}x_{n-2}^{\alpha_{n-2}-j}\delta_{n-2}(\sigma_{n-2}^{j-1}(\sigma_{n-1}^{\alpha_{n-1}}(\sigma_{n}^{\alpha_{n}}(r))))x_{n-2}^{j-1}\biggr{)}x_{n-1}^{\alpha_{n-1}}x_{n}^{\alpha_{n}}$
$\displaystyle+$ $\displaystyle\
\dotsb+x_{1}^{\alpha_{1}}\biggl{(}\sum_{j=1}^{\alpha_{2}}x_{2}^{\alpha_{2}-j}\delta_{2}(\sigma_{2}^{j-1}(\sigma_{3}^{\alpha_{3}}(\sigma_{4}^{\alpha_{4}}(\dotsb(\sigma_{n}^{\alpha_{n}}(r))))))x_{2}^{j-1}\biggr{)}x_{3}^{\alpha_{3}}x_{4}^{\alpha_{4}}\dotsb
x_{n-1}^{\alpha_{n-1}}x_{n}^{\alpha_{n}}$ $\displaystyle+$ $\displaystyle\
\sigma_{1}^{\alpha_{1}}(\sigma_{2}^{\alpha_{2}}(\dotsb(\sigma_{n}^{\alpha_{n}}(r))))x_{1}^{\alpha_{1}}\dotsb
x_{n}^{\alpha_{n}},\ \ \ \ \ \ \ \ \ \ \sigma_{j}^{0}:={\rm id}_{R}\ \ {\rm
for}\ \ 1\leq j\leq n.$

If $a_{i},b_{j}\in R$ and $X_{i}:=x_{1}^{\alpha_{i1}}\dotsb
x_{n}^{\alpha_{in}},\ Y_{j}:=x_{1}^{\beta_{j1}}\dotsb x_{n}^{\beta_{jn}}$,
when we compute every summand of $a_{i}X_{i}b_{j}Y_{j}$ we obtain products of
the coefficient $a_{i}$ with several evaluations of $b_{j}$ in $\sigma$’s and
$\delta$’s depending on the coordinates of $\alpha_{i}$. This assertion
follows from the expression:

$\displaystyle a_{i}X_{i}b_{j}Y_{j}=$ $\displaystyle\
a_{i}\sigma^{\alpha_{i}}(b_{j})x^{\alpha_{i}}x^{\beta_{j}}+a_{i}p_{\alpha_{i1},\sigma_{i2}^{\alpha_{i2}}(\dotsb(\sigma_{in}^{\alpha_{in}}(b_{j})))}x_{2}^{\alpha_{i2}}\dotsb
x_{n}^{\alpha_{in}}x^{\beta_{j}}$ $\displaystyle+$ $\displaystyle\
a_{i}x_{1}^{\alpha_{i1}}p_{\alpha_{i2},\sigma_{3}^{\alpha_{i3}}(\dotsb(\sigma_{{in}}^{\alpha_{in}}(b_{j})))}x_{3}^{\alpha_{i3}}\dotsb
x_{n}^{\alpha_{in}}x^{\beta_{j}}$ $\displaystyle+$ $\displaystyle\
a_{i}x_{1}^{\alpha_{i1}}x_{2}^{\alpha_{i2}}p_{\alpha_{i3},\sigma_{i4}^{\alpha_{i4}}(\dotsb(\sigma_{in}^{\alpha_{in}}(b_{j})))}x_{4}^{\alpha_{i4}}\dotsb
x_{n}^{\alpha_{in}}x^{\beta_{j}}$ $\displaystyle+$ $\displaystyle\
\dotsb+a_{i}x_{1}^{\alpha_{i1}}x_{2}^{\alpha_{i2}}\dotsb
x_{i(n-2)}^{\alpha_{i(n-2)}}p_{\alpha_{i(n-1)},\sigma_{in}^{\alpha_{in}}(b_{j})}x_{n}^{\alpha_{in}}x^{\beta_{j}}$
$\displaystyle+$ $\displaystyle\ a_{i}x_{1}^{\alpha_{i1}}\dotsb
x_{i(n-1)}^{\alpha_{i(n-1)}}p_{\alpha_{in},b_{j}}x^{\beta_{j}},$

where the polynomials $p$’s are given by Proposition 2.4.

###  2.1. $(\Sigma,\Delta)$-compatible and weak $(\Sigma,\Delta)$-compatible
rings

As we said in the Introduction, following Annin [3] (see also Hashemi and
Moussavi [45]), for an endomorphism $\sigma$ and a $\sigma$-derivation
$\delta$ of a ring $R$, $R$ is said to be $\sigma$-compatible if for every
$a,b\in R$, we have $ab=0$ if and only if $a\sigma(b)=0$ (necessarily,
$\sigma$ is injective). $R$ is called $\delta$-compatible if for each $a,b\in
R$, $ab=0$ implies $a\delta(b)=0$. If $R$ is both $\sigma$-compatible and
$\delta$-compatible, then $R$ is said to be $(\sigma,\delta)$-compatible. From
[44, Lemma 3.3] we know that $(\sigma,\delta)$-compatible rings generalize the
$\sigma$-rigid rings (an endomorphism $\sigma$ of a ring $R$ is called rigid
if $r\sigma(r)=0$ implies $r=0$, where $r\in R$; $R$ is said to be rigid if
there exists a rigid endomorphism $\sigma$ of $R$) introduced by Krempa [64].
Thinking in the context of skew PBW extensions, Hashemi et al. [41] and Reyes
and Suárez [94] introduced independently the $(\Sigma,\Delta)$-compatible
rings as a natural generalization of $(\sigma,\delta)$-compatible rings.
Examples, ring and module theoretic properties of these structures can be
found in [40, 42, 43, 73, 91, 97].

Since the notion of $(\sigma,\delta)$-compatible ring was used by Ouyang and
Birkenmeier [82] in their treatment about weak annihilators of Ore extensions,
it is to be expected that we have to consider the notion of
$(\Sigma,\Delta)$-compatible ring to characterize this type of annihilators in
the setting of skew PBW extensions. In this way, next we consider a finite
family of endomorphisms $\Sigma$ and a finite family $\Delta$ of
$\Sigma$-derivations of a ring $R$.

######  Definition 2.6 ([41, Definition 3.1]; [94, Definition 3.2]).

$R$ is said to be $\Sigma$-compatible if for each $a,b\in R$,
$a\sigma^{\alpha}(b)=0$ if and only if $ab=0$, where
$\alpha\in\mathbb{N}^{n}$. $R$ is said to be $\Delta$-compatible if for each
$a,b\in R$, it follows that $ab=0$ implies $a\delta^{\beta}(b)=0$, where
$\beta\in\mathbb{N}^{n}$. If $R$ is both $\Sigma$-compatible and
$\Delta$-compatible, then $R$ is called $(\Sigma,\Delta)$-compatible.

The following proposition is the natural generalization of [45, Lemma 2.1].

######  Proposition 2.7 ([41, Lemma 3.3]; [94, Proposition 3.8]).

If $R$ is a $(\Sigma,\Delta)$-compatible ring, then for every elements $a,b$
belong to $R$, we have the following assertions:

  1. (1)

If $ab=0$, then $a\sigma^{\theta}(b)=\sigma^{\theta}(a)b=0$, where
$\theta\in\mathbb{N}^{n}$.

  2. (2)

If $\sigma^{\beta}(a)b=0$, for some $\beta\in\mathbb{N}^{n}$, then $ab=0$.

  3. (3)

If $ab=0$, then
$\sigma^{\theta}(a)\delta^{\beta}(b)=\delta^{\beta}(a)\sigma^{\theta}(b)=0$,
where $\theta,\beta\in\mathbb{N}^{n}$.

######  Example 2.8.

Let $\mathbb{F}_{4}=\left\\{0,1,a,a^{2}\right\\}$ be the field of four
elements. Consider the ring of polynomials $\mathbb{F}_{4}[z]$ and let
$R=\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}$. For simplicity,
we identify the elements of $\mathbb{F}_{4}[z]$ with their images in $R$. Let
$\Sigma=\left\\{\sigma_{i,j}\right\\}$ be the family of homomorphisms of $R$
defined by $\sigma_{i,j}(a)=a^{i}$ and $\sigma_{i,j}(z)=a^{j}z$, for $1\leq
i\leq 2$ and $0\leq j\leq 2$. Consider the skew PBW extension
$A=\sigma(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$, where the
indeterminates satisfy the commutation relations
$x_{i,j}x_{i^{\prime},j^{\prime}}=x_{i^{\prime},j^{\prime}}x_{i,j}$, for all
$1\leq i,i^{\prime}\leq 2$ and $0\leq j,j^{\prime}\leq 2$. On the other hand,
for $a^{r}z\in R$, we have that
$x_{i,j}a^{r}z=\sigma_{i,j}(a^{r}z)x_{i,j}=(a^{r})^{i}a^{j}zx_{i,j}=a^{ri+j}zx_{i,j}$,
where $a^{ri+j}\in\mathbb{F}_{4}$, $1\leq i\leq 2$ and $1\leq r,j\leq 2$. This
example can be extended to any finite field $\mathbb{F}_{p^{n}}$ with $p$ a
prime number. It is straightforward to show that $R$ is $\Sigma$-compatible,
and thus $A$ is a skew PBW extension over a $\Sigma$-compatible ring $R$.

######  Example 2.9.

Let $\Bbbk[t]$ be the commutative polynomial ring over the field $\Bbbk$ in
the indeterminate $t$. We consider the identity homomorphism $\sigma$ of
$\Bbbk[t]$ and $\delta(t):=1$. Having in mind that $\Bbbk[t]$ is reduced, we
can show that $\Bbbk[t]$ is a $(\sigma,\delta)$-compatible ring. Let $R$ be
the ring defined as

$\displaystyle R=\left\\{\begin{pmatrix}p(t)&q(t)\\\ 0&p(t)\end{pmatrix}\ \mid
p(t),q(t)\in\Bbbk[t]\right\\}$.

The endomorphism $\sigma$ of $\Bbbk[t]$ is extended to the endomorphism
$\overline{\sigma}:R\rightarrow R$ by defining
$\overline{\sigma}((a_{ij}))=(\sigma(a_{ij}))$, and the $\sigma$-derivation
$\delta$ of $\Bbbk[t]$ is also extended to $\overline{\delta}:R\rightarrow R$
by considering $\overline{\delta}((a_{ij}))=(\delta(a_{ij}))$. Notice that the
Ore extension $R[x;\overline{\sigma},\overline{\delta}]$ is a skew PBW
extension over $R$ which is
$(\overline{\sigma},\overline{\delta})$-compatible.

The following definition presents the notion of compatibility for ideals of a
ring.

######  Definition 2.10 ([42, Definition 3.1]).

Let $R$ be a $(\Sigma,\Delta)$-compatible ring where, as above,
$\Sigma:=\left\\{\sigma_{1},\dots,\sigma_{n}\right\\}$ is a finite family of
endomorphisms of $R$ and
$\Delta:=\left\\{\delta_{1},\dots,\delta_{n}\right\\}$ is a finite family of
$\Sigma$-derivations of $R$.

  1. (i)

An ideal $I$ of $R$ is called a $\Sigma$-ideal if $\sigma^{\alpha}(I)\subseteq
I$, where $\alpha\in\mathbb{N}^{n}$. $I$ is a $\Delta$-ideal if
$\delta^{\alpha}(I)\subseteq I$, where $\alpha\in\mathbb{N}^{n}$. $I$ is
$\Sigma$-invariant if $\sigma^{-\alpha}(I)=I$, where
$\alpha\in\mathbb{N}^{n}$. If $I$ is both $\Sigma$ and $\Delta$-ideal, then
$I$ is called a $(\Sigma,\Delta)$-ideal.

  2. (ii)

For an ideal $I$ of $R$, we say that $I$ is $\Sigma$-compatible if for each
$a,b\in R$ and $\alpha\in\mathbb{N}^{n}$, $ab\in I$ if and only if
$a\sigma^{\alpha}(b)\in I$. Moreover, $I$ is said to be $\Delta$-compatible
ideal if for each $a,b\in R$ and $\alpha\in\mathbb{N}^{n}$, $ab\in I$ implies
$a\delta^{\alpha}(b)\in I$. If $I$ is both $\Sigma$-compatible and
$\Delta$-compatible, then we say that $I$ is $(\Sigma,\Delta)$-compatible.

Notice that $R$ is $\Sigma$-compatible (resp., $\Delta$-compatible) if and
only if the zero ideal is $\Sigma$-compatible (resp., $\Delta$-compatible)
ideal of $R$. If $R$ is a $(\Sigma,\Delta)$-compatible ring and $N(R)$ is an
ideal of $R$, then, $N(R)$ is a $(\Sigma,\Delta)$-compatible ideal.

######  Proposition 2.11 ([42, Proposition 3.3]).

If $I$ is a $(\Sigma,\Delta)$-compatible ideal of $R$ and $a,b$ are elements
of $R$, then the following assertions hold:

  1. (1)

If $ab\in I$, then $a\sigma^{\alpha}(b)\in I$ and $\sigma^{\alpha}(a)b\in I$,
for each $\alpha\in\mathbb{N}^{n}$.

  2. (2)

If $ab\in I$, then $\sigma^{\alpha}(a)\delta^{\beta}(b),\
\delta^{\beta}(a)\sigma^{\alpha}(b)\in I$, for every
$\alpha,\beta\in\mathbb{N}^{n}$.

  3. (3)

If $a\sigma^{\theta}(b)\in I$ or $\sigma^{\theta}(a)b\in I$, for some
$\theta\in\mathbb{N}^{n}$, then $ab\in I$.

Reyes and Suárez [95] defined the concept of weak $(\Sigma,\Delta)$-compatible
ring as a generalization of $(\Sigma,\Delta)$-compatible rings and weak
$(\sigma,\delta)$-compatible rings introduced by Ouyang and Liu [83]. For the
next definition, consider again a finite family of endomorphisms $\Sigma$ and
a finite family $\Delta$ of $\Sigma$-derivations of a ring $R$.

######  Definition 2.12 ([95, Definition 4.1]).

$R$ is said to be weak $\Sigma$-compatible if for each $a,b\in R$,
$a\sigma^{\alpha}(b)\in N(R)$ if and only if $ab\in N(R)$, where
$\alpha\in\mathbb{N}^{n}$. $R$ is said to be weak $\Delta$-compatible if for
each $a,b\in R$, $ab\in N(R)$ implies $a\delta^{\beta}(b)\in N(R)$, where
$\beta\in\mathbb{N}^{n}$. If $R$ is both weak $\Sigma$-compatible and weak
$\Delta$-compatible, then $R$ is called weak $(\Sigma,\Delta)$-compatible.

It is clear that the following result extends Proposition 2.7.

######  Proposition 2.13 ([95, Proposition 4.2]).

If $R$ is a weak $(\Sigma,\Delta)$-compatible ring, then the following
assertions hold:

  * (1)

If $ab\in N(R)$, then $a\sigma^{\alpha}(b),\sigma^{\beta}(a)b\in N(R)$, for
all $\alpha,\beta\in\mathbb{N}^{n}$.

  * (2)

If $\sigma^{\alpha}(a)b\in N(R)$, for some element $\alpha\in\mathbb{N}^{n}$,
then $ab\in N(R)$.

  * (3)

If $a\sigma^{\beta}(b)\in N(R)$, for some element $\beta\in\mathbb{N}^{n}$,
then $ab\in N(R)$.

  * (4)

If $ab\in N(R)$, then
$\sigma^{\alpha}(a)\delta^{\beta}(b),\delta^{\beta}(a)\sigma^{\alpha}(b)\in
N(R)$, for $\alpha,\beta\in\mathbb{N}^{n}$.

The next example shows that there exists a weak $(\Sigma,\Delta)$-compatible
ring which is not $(\Sigma,\Delta)$-compatible.

######  Example 2.14.

Let $R$ be a ring and $M$ an $(R,R)$-bimodule. The trivial extension of R by M
is defined as the ring $T(R,M):=R\oplus M$ with the usual addition and the
multiplication defined as
$(r_{1},m_{1})(r_{2},m_{2}):=(r_{1}r_{2},r_{1}m_{2}+m_{1}r_{2})$, for all
$r_{1},r_{2}\in R$ and $m_{1},m_{2}\in M$. Notice that $T(R,M)$ is isomorphic
to the matrix ring (with the usual matrix operations) of the form
$\bigl{(}\begin{smallmatrix}r&m\\\ 0&r\end{smallmatrix}\bigr{)}$, where $r\in
R$ and $m\in M$. In particular, we call ${\rm S}_{2}(\mathbb{Z})$ the ring of
matrices isomorphic to $T(\mathbb{Z},\mathbb{Z})$.

Consider the ring ${\rm S}_{2}(\mathbb{Z})$ given by

${\rm S}_{2}(\mathbb{Z})=\left\\{\begin{pmatrix}a&b\\\ 0&a\end{pmatrix}\mid
a,b\in\mathbb{Z}\right\\}$.

Let $\sigma_{1}={\rm id}_{{\rm S}_{2}(\mathbb{Z})}$ be the identity
endomorphism of ${\rm S}_{2}(\mathbb{Z})$, and let $\sigma_{2}$ and
$\sigma_{3}$ be the two endomorphisms of ${\rm S}_{2}(\mathbb{Z})$ defined by

$\sigma_{2}\left(\begin{pmatrix}a&b\\\
0&a\end{pmatrix}\right)=\begin{pmatrix}a&-b\\\ 0&a\end{pmatrix},\quad{\rm
and}\quad\sigma_{3}\left(\begin{pmatrix}a&b\\\
0&a\end{pmatrix}\right)=\begin{pmatrix}a&0\\\ 0&a\end{pmatrix}$,

respectively. Notice that the set of nilpotent elements of ${\rm
S}_{2}(\mathbb{Z})$ is given by

$N({\rm S}_{2}(\mathbb{Z}))=\left\\{\begin{pmatrix}0&b\\\ 0&0\end{pmatrix}\mid
b\in\mathbb{Z}\right\\}$.

It is straightforward to see that ${\rm S}_{2}(\mathbb{Z})$ is weak
$\Sigma$-compatible where
$\Sigma=\left\\{\sigma_{1},\sigma_{2},\sigma_{3}\right\\}$, but ${\rm
S}_{2}(\mathbb{Z})$ is not $\sigma_{3}$-compatible. For example, if we take
the matrices $C,D\in{\rm S}_{2}(\mathbb{Z})$ given by

$C=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\ \ D=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},$

then it is clear that $C\sigma_{3}(D)=0$ but $CD=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\neq 0$, whence ${\rm S}_{2}(\mathbb{Z})$ is not
$\Sigma$-compatible.

Ouyang and Liu [83, Lemma 2.13] characterized the nilpotent elements in an Ore
extension over a weak $(\sigma,\delta)$-compatible and NI ring. Since skew PBW
extensions generalize Ore extensions of injective type, Reyes and Suárez [95]
extended this result as the following proposition shows. The NI property for
skew PBW extensions was recently studied by Suárez et al. [104].

######  Proposition 2.15 ([95, Theorem 4.6]).

If $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ is a skew PBW extension over
a weak $(\Sigma,\Delta)$-compatible and NI ring $R$, then
$f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}\in N(A)$ if and only if $a_{i}\in N(R)$,
for every $i$.

Notice that Proposition 2.15 also generalizes the results presented by Ouyang
and Birkenmeier for Ore extensions [82, Lemma 2.6], and Ouyang and Liu for
differential polynomials rings [84, Lemma 2.12]. Furthermore, since weak
$(\Sigma,\Delta)$-compatible rings contain strictly
$(\Sigma,\Delta)$-compatible rings, we have immediately the following
corollary for skew PBW extensions over $(\Sigma,\Delta)$-compatible rings.

######  Corollary 2.16.

If $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ is a skew PBW extension over
a $(\Sigma,\Delta)$-compatible and NI ring $R$, then the following statements
hold:

  1. (1)

$N(A)$ is an ideal of $A$ and $N(A)=N(R)\langle x_{1},\dotsc,x_{n}\rangle$.

  2. (2)

$f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}\in N(A)$ if and only if $a_{i}\in N(R)$,
for every $i=0,\dotsc,m$.

Corollary 2.16 generalizes the corresponding theorem for Ore extensions of
injective type over $(\sigma,\delta)$-compatible and $2$-primal rings [82,
Corollary 2.2], and the result for differential polynomials rings over
$\delta$-compatible reversible rings [84, Corollary 2.13].

Ouyang et al. [85] introduced the notion of skew $\pi$-Armendariz ring in the
following way. If $R$ is a ring with an endomorphism $\sigma$ and a
$\sigma$-derivation $\delta$, then $R$ is called skew $\pi$-Armendariz ring if
for polynomials $f(x)=\sum_{i=0}^{l}a_{i}x^{i}$ and
$g(x)=\sum_{j=0}^{m}b_{j}x^{j}$ in $R[x;\sigma,\delta]$, $f(x)g(x)\in
N(R[x;\sigma,\delta])$ implies that $a_{i}b_{j}\in N(R)$, for each $0\leq
i\leq l$ and $0\leq j\leq m$. Skew $\pi$-Armendariz rings are more general
than skew Armendariz rings when the ring of coefficients is
$(\sigma,\delta)$-compatible [85, Theorem 2.6], and also extend
$\sigma$-Armendariz rings defined by Hong et al. [47] considering $\delta$ as
the zero derivation. For a detailed description of Armendariz rings in the
commutative and noncommutative context, including skew PBW extensions, see [7,
22, 50].

Ouyang and Liu [83] showed that if $R$ is a weak $(\sigma,\delta)$-compatible
and NI ring, then $R$ is skew $\pi$-Armendariz ring [83, Corollary 2.15].
Reyes [90] formulated the analogue of skew $\pi$-Armendariz ring in the
setting of skew PBW extensions. For a skew PBW extension $A=\sigma(R)\langle
x_{1},\dotsc,x_{n}\rangle$ over a ring $R$, $R$ is said to be skew
$\Pi$-Armendariz ring if for elements $f=\sum_{i=0}^{l}a_{i}X_{i}$ and
$g=\sum_{j=0}^{m}b_{j}Y_{j}$ belong to $A$, $fg\in N(A)$ implies
$a_{i}b_{j}\in N(R)$, for each $0\leq i\leq l$ and $0\leq j\leq m$. Reyes [90,
Theorem 3.10] showed that if $R$ is reversible (following Cohn [27, p. 641],
$R$ is called reversible if $ab=0$ implies $ba=0$, where $a,b\in R$) and
$(\Sigma,\Delta)$-compatible, then $R$ is skew $\Pi$-Armendariz ring. Related
to this result, the following proposition generalizes [84, Lemma 2.14] and
[82, Corollary 2.3].

######  Proposition 2.17 ([95, Theorem 4.7]).

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a skew PBW extension
over a weak $(\Sigma,\Delta)$-compatible and NI ring $R$. If
$f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}\in N(A)$ and
$g=b_{0}+b_{1}Y_{1}+\dotsb+b_{t}Y_{t}$ are elements of $A$, then $fg\in N(A)$
if and only if $a_{i}b_{j}\in N(R)$, for all $i,j$.

##  3\. Weak Annihilator Ideals

In this section, we study the weak notion of annihilator introduced by Ouyang
and Birkenmeier [82] in the setting of skew PBW extensions.

######  Definition 3.1 ([82, Definition 2.1]).

Let $R$ be a ring. For $X\subseteq R$, it is defined $N_{R}(X)=\left\\{a\in R\
|\ xa\in N(R)\ \text{for all}\ x\in X\right\\}$, which is called the weak
annihilator of $X$ in $R$. If $X$ is a singleton, say $X=\left\\{r\right\\}$,
we use $N_{R}(r)$ to denote $N_{R}(\left\\{r\right\\})$.

Notice that for $X\subseteq R$, the sets given by $\left\\{a\in R\ |\ xa\in
N(R)\ \text{for all}\ x\in X\right\\}$ and $\left\\{b\in R\ |\ bx\in N(R)\
\text{for all}\ x\in X\right\\}$ coincide. Moreover, $l_{R}(X),\
r_{R}(X)\subseteq N_{R}(X)$. It is clear that if $R$ is reduced, then
$r_{R}(X)=l_{R}(X)=N_{R}(X)$. In addition, if $N(R)$ is an ideal of $R$, then
$N_{R}(X)$ is also an ideal of $R$ [82, p. 346].

######  Example 3.2 ([82, Example 2.1]).

Let $T_{2}(\mathbb{Z})$ be the $2\times 2$ upper triangular matrix ring over
$\mathbb{Z}$ and $X=\left\\{\bigl{(}\begin{smallmatrix}2&0\\\
0&2\end{smallmatrix}\bigr{)}\right\\}\subseteq T_{2}(\mathbb{Z})$. Then
$r_{T_{2}(\mathbb{Z})}(X)=0$ and
$N_{T_{2}(\mathbb{Z})}(X)=\left\\{\bigl{(}\begin{smallmatrix}0&m\\\
0&0\end{smallmatrix}\bigr{)}\ |\ m\in\mathbb{Z}\right\\}$, whence
$r_{T_{2}(\mathbb{Z})}(X)\neq N_{T_{2}(\mathbb{Z})}(X)$. This shows that a
weak annihilator is not an immediate generalization of an annihilator.

We recall some properties of weak annihilators.

######  Proposition 3.3 ([82, Proposition 2.1]).

If $X$ and $Y$ are subsets of a ring $R$, then we have the following:

  * (1)

$X\subseteq Y$ implies $N_{R}(Y)\subseteq N_{R}(X)$.

  * (2)

$X\subseteq N_{R}(N_{R}(X)))$.

  * (3)

$N_{R}(X)=N_{R}(N_{R}(N_{R}(X)))$.

For a skew PBW extension $A$ over a ring $R$, let ${\rm
NAnn}_{R}(R):=\left\\{N_{R}(U)\ |\ U\subseteq R\right\\}$ and ${\rm
NAnn}_{A}(A):=\left\\{N_{A}(V)\ |\ V\subseteq A\right\\}$. For
$f=a_{0}+a_{1}X_{1}+\cdots+a_{m}X_{m}\in A$, we denote by
$\left\\{a_{0},\ldots,a_{m}\right\\}$ or $C_{f}$ the set comprised of the
coefficients of $f$, and for a subset $V\subseteq A$, $C_{V}=\bigcup_{f\in
U}C_{f}$.

The following theorem establishes a bijective correspondence between weak
annihilators of $A$ and weak annihilators of $R$. This result generalizes [96,
Theorem 3.21] which was formulated for classical annihilators.

######  Theorem 3.4.

If $A=\sigma(R)\left\langle x_{1},\dotsc,x_{n}\right\rangle$ is a skew PBW
extension over a $(\Sigma,\Delta)$-compatible and NI ring $R$, then the
correspondence $\varphi:{\rm NAnn}_{R}(R)\rightarrow{\rm NAnn}_{A}(A)$, given
by $\varphi(N_{R}(U))=N_{R}(U)A$, for every $N_{R}(U)\in{\rm NAnn}_{R}(R)$, is
bijective.

###### Proof.

First, let us show that $\varphi$ is well defined, i.e., $N_{A}(U)=N_{R}(U)A$,
for every nonempty subset $U$ of $R$. Consider
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N_{R}(U)A$ and $r\in U$. Since
$a_{i}\in N_{R}(U)$, for all $0\leq i\leq l$, this implies that $ra_{i}\in
N(R)$, for each $i$. Thus, by Corollary (1), we get
$rf=ra_{0}+ra_{1}X_{1}+\cdots+ra_{l}X_{l}\in N(R)A=N(A)$. Hence,
$N_{R}(U)A\subseteq N_{A}(U)$.

Now, let us prove that $N_{A}(U)\subseteq N_{R}(U)A$. If
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N_{A}(U)$, then
$rf=ra_{0}+ra_{1}X_{1}+\cdots+ra_{l}X_{l}\in N(A)=N(R)A$, for every element
$r$ of $U$. This means that $ra_{i}\in N(R)$, for all $0\leq i\leq l$, and so
$a_{i}\in N_{R}(U)$, for each $i$, whence $f\in N_{R}(U)A$. Therefore,
$N_{A}(U)=N_{R}(U)A$.

Let $U,V$ be a two subsets of $R$ such that
$\varphi(N_{R}(U))=\varphi(N_{R}(V))$. By definition of $\varphi$,
$N_{A}(U)=N_{A}(V)$. In particular, we obtain $N_{R}(U)=N_{R}(V)$, that is,
$\varphi$ is injective.

Finally, let us show that $\varphi$ is surjective. Let $N_{A}(V)\in{\rm
NAnn}_{A}(A)$ and consider $g=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}\in N_{A}(V)$,
for a subset $V$ of $A$. Then $fg\in N(A)$, for all
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in V$, and by Proposition 2.17,
$a_{i}b_{j}\in N(R)$, for each $i,j$. Thus, $b_{j}\in N_{R}(C_{V})$, for all
$0\leq j\leq s$, whence $g\in N_{R}(C_{V})A$, and so $N_{A}(V)\subseteq
N_{R}(C_{V})A$. Since $N_{R}(C_{V})A\subseteq N_{A}(V)$, we obtain that
$N_{A}(V)=N_{R}(C_{V})A=\varphi(N_{R}(C_{V}))$, i.e., $\varphi$ is surjective.
∎

The following theorem generalizes [82, Theorem 2.1].

######  Theorem 3.5.

Let $A=\sigma(R)\left\langle x_{1},\dots,x_{n}\right\rangle$ be a skew PBW
extension over a $(\Sigma,\Delta)$-compatible and NI ring $R$. If for each
subset $X\nsubseteq N(R)$, $N_{R}(X)$ is generated as an ideal by a nilpotent
element, then for each subset $U\nsubseteq N(A)$, $N_{A}(U)$ is generated as
an ideal by a nilpotent element.

###### Proof.

Let $U$ be a subset of $A$ with $U\nsubseteq N(A)$. By Corollary (2),
$C_{U}\nsubseteq N(R)$. There exists $c\in N(R)$ such that $N_{R}(C_{U})=cR$.
Let us show that $N_{A}(U)=cA$. Let $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in
U$ and $g=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}\in A$. The idea is to show that
$fcg\in N(A)$. Using that $cg=cb_{0}+cb_{1}Y_{1}+\cdots+cb_{s}Y_{s}$, we get

$\displaystyle
fcg=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}cb_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(cb_{j})X_{i}Y_{j}+a_{i}p_{\alpha_{i},cb_{j}}Y_{j}\right)$.

Since $a_{i}cb_{j}\in N(R)$, for every $0\leq i\leq l$, $0\leq j\leq s$, [42,
Proposition 3.3] this implies that $a_{i}\sigma^{\alpha_{i}}(cb_{j})\in N(R)$.
Notice that $a_{i}\sigma^{\alpha}(\delta^{\beta}(cb_{j}))$ and
$a_{i}\delta^{\beta}(\sigma^{\alpha}(cb_{j}))$ are elements of $N(R)$, for
every $\alpha,\beta\in\mathbb{N}^{n}$. In addition, by Proposition 2.5, the
polynomial $p_{\alpha_{i},cb_{j}}$ involves elements obtained evaluating
$\sigma$’s and $\delta$’s (depending on the coordinates of $\alpha_{i}$) in
the element $cb_{j}$. Thus, $a_{i}p_{\alpha_{i},cb_{j}}\in N(R)$ and
$a_{i}cb_{j}\in N(R)$, for every $i,j$, whence $fcg\in N(R)A=N(A)$, and
therefore $cg\in N_{A}(U)$.

Now, let us consider $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N_{A}(U)$. By
definition, for all $g=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}$ it is satisfied
that $fg\in N(A)$. From Corollary (2), $fg\in N(R)A$, and so

$\displaystyle
fg=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}b_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(b_{j})X_{i}Y_{j}+p_{\alpha_{i},b_{j}}Y_{j}\right)\in
N(R)A$.

Again, since $a_{i}\sigma^{\alpha_{i}}(b_{j})\in N(R)$, for every $i,j$, [42,
Proposition 3.3] implies that $a_{i}b_{j}\in N(R)$. Hence, $a_{i}\in
N_{R}(C_{U})=cR$, for each $0\leq i\leq l$. This means that there exist
elements $r_{i}\in R$ such that $a_{i}=cr_{i}$, for every $i$. In this way,
$f=c(r_{0}+r_{1}X_{1}+\cdots+r_{l}X_{l})\in cA$, that is, $f\in cA$. ∎

######  Corollary 3.6 ([82, Theorem 2.1]).

Let $R$ be a $(\sigma,\delta)$-compatible ring $R$. If for each subset
$X\nsubseteq N(R)$, $N_{R}(X)$ is generated as an ideal by a nilpotent
element, then for each subset $U\nsubseteq N(R[x;\sigma,\delta])$,
$N_{R[x;\sigma,\delta]}(U)$ is generated as an ideal by a nilpotent element.

######  Example 3.7.

  1. (i)

Let $A$ be the skew PBW extension over the ring
$\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}$ formulated in
Example 2.8. For any subset $X\subseteq\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}$ with $X\nsubseteq
N\left(\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}\right)$, we
obtain that $N_{\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}}(X)=z\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}$, where $z\in N\left(\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\right)$. From Theorem 3.5, we conclude that $N_{A}U=wA$,
for any subset $U\subseteq A$ where $U\nsubseteq N(A)$, $w\in N(A)$ and
$A=\sigma(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$.

  2. (ii)

Consider Example 2.9. For any subset $X\subseteq R$ with $X\nsubseteq N(R)$ it
is satisfied that $N_{R}(X)=rR$, where $r\in N(R)$. Thus, by Theorem 3.5 we
have that
$N_{R[x;\overline{\sigma},\overline{\delta}]}U=wR[x;\overline{\sigma},\overline{\delta}]$,
for any subset $U\subseteq R[x;\overline{\sigma},\overline{\delta}]$, where
$U\nsubseteq N(R[x;\overline{\sigma},\overline{\delta}])$ and $w\in
N(R[x;\overline{\sigma},\overline{\delta}])$.

The following theorem is a consequence of Theorem 3.5 considering skew PBW
extensions of the endomorphism type. This result generalizes [82, Theorem
2.2].

######  Theorem 3.8.

If $A=\sigma(R)\left\langle x_{1},\dots,x_{n}\right\rangle$ is a skew PBW
extension of endomorphism type over a $\Sigma$-compatible and NI ring $R$,
then the following statements are equivalent:

  1. (1)

For each subset $X\nsubseteq N(R)$, $N_{R}(X)$ is generated as an ideal by a
nilpotent element.

  2. (2)

For each subset $U\nsubseteq N(A)$, $N_{A}(U)$ is generated as an ideal by a
nilpotent element.

###### Proof.

By Theorem 3.5, it suffices to show (2) $\Rightarrow$ (1). Let $X$ be a subset
of $R$ with $X\nsubseteq N(R)$. Then $X\nsubseteq N(A)$, and so there exists
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N(A)$ such that $N_{A}(X)=fA$. Notice
that $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N(A)$, whence $a_{i}\in N(R)$,
for all $0\leq i\leq l$, by Corollary (2). Suppose that $a_{0}\neq 0$, and let
us see that $N_{R}(X)=a_{0}R$. Since $a_{0}\in N(R)$ and $N(R)$ is an ideal of
$R$, we obtain $pa_{0}R\subseteq N(R)$, for each $p\in X$, whence
$a_{0}R\subseteq N_{R}(X)$. If $m\in N_{R}(X)$, then $m\in N_{A}(X)$. Thus,
there exists $g(x)=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}\in A$ such that

$\displaystyle
m=fg=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}b_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(b_{j})X_{i}Y_{j}\right)$

Then $m=a_{0}b_{0}\in a_{0}R$, and thus $N_{R}(X)\subseteq a_{0}R$, which
means that $N_{R}(X)=a_{0}R$, with $a_{0}\in N(R)$. ∎

######  Corollary 3.9 ([82, Theorem 2.2]).

Let $R$ be a $(\sigma,\delta)$-compatible ring $R$. Then the following
statements are equivalent:

  1. (1)

For each subset $X\nsubseteq N(R)$, $N_{R}(X)$ is generated as an ideal by a
nilpotent element.

  2. (2)

For each subset $U\nsubseteq N(R[x;\sigma])$, $N_{R[x;\sigma]}(U)$ is
generated as an ideal by a nilpotent element.

Next, we present a theorem for weak annihilators of principal right ideals.

######  Theorem 3.10.

Let $A=\sigma(R)\left\langle x_{1},\dots,x_{n}\right\rangle$ be a skew PBW
extension over a $(\Sigma,\Delta)$-compatible and NI ring $R$. If for each
principal right ideal $pR\nsubseteq N(R)$, $N_{R}(pR)$ is generated as an
ideal by a nilpotent element, then for each principal right ideal
$fA\nsubseteq N(A)$, $N_{A}(fA)$ is generated as an ideal by a nilpotent
element.

###### Proof.

Let $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in A$ with $fA\nsubseteq N(A)$. If
$a_{i}R\subseteq N(R)$, for all $0\leq i\leq l$, by Corollary (2), we have
that $fA\subseteq N(A)$, which is a contradiction. In this way, there exists
$0\leq i\leq l$ such that $a_{i}R\nsubseteq N(R)$. Thus, there exists $c\in
N(R)$ such that $N_{R}(a_{i}R)=cR$. Let us show that $N_{A}(fA)=cA$. Consider
$g=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}$ and
$h=c_{0}+c_{1}Z_{1}+\cdots+c_{t}Z_{t}$ elements of $A$. By Theorem 3.5,
$gch\in N(R)A=N(A)$ and by Corollary (2), for every $f\in A$, $fgch\in
N(A)=N(R)A$, since that $N(A)$ is an ideal. Therefore, we obtain that $ch\in
N_{A}(fA)$, that is, $cA\subseteq N_{A}(fA)$. Let
$p=p_{0}+p_{1}Y_{1}+\cdots+p_{s}Y_{s}\in N_{A}(fA)$. Then $fAp\subseteq N(A)$,
for every $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in A$. In particular,
$fRp\subseteq N(A)$. Let $r\in R$. We get
$rp=rp_{0}+rp_{1}Y_{1}\cdots+rp_{s}Y_{s}$, which implies that

$\displaystyle
frp=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}rp_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(rp_{j})X_{i}Y_{j}+a_{i}p_{\alpha_{i},rp_{j}}Y_{j}\right)\in
N(R)A$.

Therefore, $a_{i}\sigma^{\alpha_{i}}(rp_{j})\in N(R)$ and so $a_{i}rp_{j}\in
N(R)$, for every $i,j$. In particular, we obtain $p_{j}\in N_{R}(a_{i}R)=cR$,
and thus there exists $r_{j}\in R$ such that $p_{j}=cr_{j}$ Hence,
$p=p_{0}+p_{1}Y_{1}+\cdots+p_{s}Y_{s}=c(r_{0}+r_{1}Y_{1}+\cdots+r_{s}Y_{s})\in
cA$. Therefore, we conclude that $p\in cA$ and so $N_{A}(fA)\subseteq cA$. ∎

######  Corollary 3.11 ([82, Theorem 2.3]).

Let $R$ be a $(\sigma,\delta)$-compatible 2-primal ring. If for each principal
right ideal $pR\nsubseteq N(R)$, $N_{R}(pR)$ is generated as an ideal by a
nilpotent element, then for each principal right ideal
$f(x)R[x;\sigma,\delta]\nsubseteq N(R[x;\sigma,\delta])$,
$N_{R[x;\sigma,\delta]}(f(x)R[x;\sigma,\delta])$ is generated as an ideal by a
nilpotent element.

######  Example 3.12.

  1. (i)

Once more again, let $A$ be the skew PBW extension over the ring
$\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}$ considered in
Example 2.8. Let $p\in\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}$ such that $p\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\nsubseteq N\left(\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\right)$. Notice that
$N_{\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}}\left(p\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\right)=z\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}$, where $z\in N\left(\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\right)$. By Theorem 3.10 $N_{A}(fA)$ is generated by a
nilpotent element for any principal right ideal $fA$ with $fA\nsubseteq N(A)$
and $A=\sigma(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$.

  2. (ii)

In Example 2.9, let $p\in R$ such that $pR\nsubseteq N(R)$, where
$p=\begin{pmatrix}p(t)&q(t)\\\ 0&q(t)\end{pmatrix}$, for some
$p(t),q(t)\in\Bbbk[t]$ and $p(t)\neq 0$. It is straightforward to see that

$N_{R}(pR)=\left\\{\begin{pmatrix}0&q(t)\\\ 0&0\end{pmatrix}\ |\
q(t)\in\Bbbk[t]\ \right\\}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}R$,

with $\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\in N(R)$. By Theorem 3.10,
$N_{R[x;\overline{\sigma},\overline{\delta}]}(fR[x;\overline{\sigma},\overline{\delta}])$
is generated by a nilpotent element for any principal right ideal
$fR[x;\overline{\sigma},\overline{\delta}]$ where
$fR[x;\overline{\sigma},\overline{\delta}]\nsubseteq
N(R[x;\overline{\sigma},\overline{\delta}])$ with $f\in
R[x;\overline{\sigma},\overline{\delta}]$.

######  Theorem 3.13.

If $A=\sigma(R)\left\langle x_{1},\dots,x_{n}\right\rangle$ is a skew PBW
extension of endomorphism type over a $\Sigma$-compatible and NI ring $R$,
then the following statements are equivalent:

  1. (1)

For each principal right ideal $pR\nsubseteq N(R)$, $N_{R}(pR)$ is generated
as an ideal by a nilpotent element.

  2. (2)

For each principal right ideal $fA\nsubseteq N(A),N_{A}(fA)$ is generated as
an ideal by a nilpotent element.

###### Proof.

By Theorem 3.10, it suffices to show ${\rm(ii)}\Rightarrow{\rm(i)}$. Let $pR$
be a principal right ideal of $R$ with $pR\nsubseteq N(R)$. In particular, we
have that $pR\nsubseteq N(A)$. Therefore, there exists
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N(A)$ such that $N_{A}(pR)=fA$. With
this in mind, notice that $f=a_{0}+a_{1}X_{1}+\dotsc+a_{l}X_{l}\in N(A)$,
whence $a_{i}\in N(R)$, for all $0\leq i\leq l$, by Corollary (2). We may
assume that $a_{0}\neq 0$. Let us show that $N_{R}(pR)=a_{0}R$. Since
$a_{0}\in N(R)$ and $N(R)$ is an ideal of $R$, we obtain $pra_{0}R\subseteq
N(R)$, for each $r\in R$. Thus $a_{0}R\subseteq N_{R}(pR)$. If $m\in
N_{R}(pR)$, then $m\in N_{A}(pR)$, and so there exists
$g=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}\in A$ such that

$\displaystyle
m=fg=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}b_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(b_{j})X_{i}Y_{j}\right)$

Hence, we have $m=a_{0}b_{0}\in a_{0}R$, whence $N_{R}(pR)\subseteq a_{0}R$.
Therefore, we conclude that $N_{R}(pR)=a_{0}R$ where $a_{0}\in N(R)$. ∎

######  Corollary 3.14 ([82, Theorem 2.4]).

Let $R$ be a $\sigma$-compatible 2-primal ring. Then the following statements
are equivalent:

  1. (1)

For each principal right ideal $pR\nsubseteq N(R)$, $N_{R}(pR)$ is generated
as an ideal by a nilpotent element.

  2. (2)

For each principal right ideal $f(x)R[x;\sigma]\nsubseteq N(R[x;\sigma])$,
$N_{R[x;\sigma]}(f(x)R[x;\sigma])$ is generated as an ideal by a nilpotent
element.

######  Theorem 3.15.

Let $A=\sigma(R)\left\langle x_{1},\dots,x_{n}\right\rangle$ be a skew PBW
extension over a $(\Sigma,\Delta)$-compatible and NI ring $R$. If for each
$p\notin N(R)$, $N_{R}(p)$ is generated as an ideal by a nilpotent element,
then for each $f\notin N(A)$, $N_{A}(f)$ is generated as an ideal by a
nilpotent element.

###### Proof.

Consider $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in A$ with $f\notin N(A)$, and
let us see that $N_{A}(f)$ is generated as an ideal by a nilpotent element. If
$a_{i}R\subseteq N(R)$, for all $0\leq i\leq l$, Corollary (2) implies that
$f\in N(A)$, which is a contradiction. Hence, there exist $0\leq i\leq l$ such
that $a_{i}\notin N(R)$, and so there is $c\in N(R)$ such that
$N_{R}(a_{i})=cR$. The idea is to show that $N_{A}(f)=cA$. If
$h=c_{0}+c_{1}Z_{1}+\cdots+c_{t}Z_{t}\in A$, by Proposition 3.5, we have
$fch\in N(R)A=N(A)$. Therefore, we obtain that $ch\in N_{A}(f)$. On the other
hand, let $p=p_{0}+p_{1}Y_{1}+\cdots+p_{s}Y_{s}\in N_{A}(f)$. For
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in A$, we have $fp\in N(A)$. Thus, we
obtain that

$\displaystyle
fp=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}p_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(p_{j})X_{i}Y_{j}+a_{i}p_{\alpha_{i},p_{j}}Y_{j}\right)\in
N(R)A$

Therefore, $a_{i}\sigma^{\alpha_{i}}(p_{j})\in N(R)$ and so $a_{i}p_{j}\in
N(R)$ for every $i,j$. In particular, we obtain that $p_{j}\in
N_{R}(a_{i})=cR$, and thus there exists $r_{j}\in R$ such that $p_{j}=cr_{j}$,
whence
$p=p_{0}+p_{1}Y_{1}+\cdots+p_{s}Y_{s}=c(r_{0}+r_{1}Y_{1}+\cdots+r_{s}Y_{s})\in
cA$. Therefore, we have proved that $p\in cA$. ∎

######  Corollary 3.16 ([82, Theorem 2.5]).

Let $R$ be a $(\sigma,\delta)$-compatible 2-primal ring. If for each $p\notin
N(R)$, $N_{R}(p)$ is generated as an ideal by a nilpotent element, then for
each $f(x)\notin N(R[x;\sigma,\delta])$, $N_{R[x;\sigma,\delta]}(f(x))$ is
generated as an ideal by a nilpotent element.

######  Example 3.17.

  1. (i)

If $A$ is the skew PBW extension over the ring
$\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}$, let
$p\in\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}$ such that
$p\notin N\left(\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\right)$. Notice that
$N_{\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}}\left(p\right)=z\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}$, where $z\in N\left(\frac{\mathbb{F}_{4}[z]}{\left\langle
z^{2}\right\rangle}\right)$. By using Theorem 3.15 we conclude that the ideal
$N_{A}(f)$ is generated by a nilpotent element for any element $f$ where
$f\notin N(A)$ and $A=\sigma(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$.

  2. (ii)

In Example 2.9, let $p\in R$ such that $p\notin N(R)$, where
$p=\begin{pmatrix}p(t)&q(t)\\\ 0&p(t)\end{pmatrix}$, for some
$p(t),q(t)\in\Bbbk[t]$ and $p(t)\neq 0$. It is easy to see that

$N_{R}(p)=\left\\{\begin{pmatrix}0&q(t)\\\ 0&0\end{pmatrix}\ |\
q(t)\in\Bbbk[t]\ \right\\}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}R$,

where $\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\in N(R)$. By using Theorem 3.15
we obtain that the ideal $N_{R[x;\overline{\sigma},\overline{\delta}]}(f)$ is
generated by a nilpotent element of
$R[x;\overline{\sigma},\overline{\delta}]$, for any element $f\in
R[x;\overline{\sigma},\overline{\delta}]$ where $f\notin
N(R[x;\overline{\sigma},\overline{\delta}])$.

For a skew PBW extension of endomorphism type over a ring $R$, the following
theorem characterizes extensions for which every weak annihilator of an
element is generated by a nilpotent element. Our result extends [82, Theorem
2.6].

######  Theorem 3.18.

If $A=\sigma(R)\left\langle x_{1},\dots,x_{n}\right\rangle$ is a skew PBW
extension of endomorphism type over a $\Sigma$-compatible and NI ring $R$,
then the following statements are equivalent:

  1. (1)

For each $p\notin N(R)$, $N_{R}(p)$ is generated as an ideal by a nilpotent
element.

  2. (2)

For each $f\notin N(A)$, $N_{A}(f)$ is generated as an ideal by a nilpotent
element.

###### Proof.

By Theorem 3.15, it suffices to show ${\rm(ii)}\Rightarrow{\rm(i)}$. Let $p\in
R$ with $p\notin N(R)$, whence $p\notin N(A)$. Thus, there exists
$f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N(A)$ such that $N_{A}(p)=fA$. Notice
that $f=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in N(A)$ and so $a_{i}\in N(R)$,
for all $0\leq i\leq l$ by Corollary (2). We may assume that $a_{0}\neq 0$.
Now, we show that $N_{R}(p)=a_{0}R$. Since $a_{0}\in N(R)$ and $N(R)$ is an
ideal of $R$, we obtain $pa_{0}R\subseteq N(R)$, that is, $a_{0}R\subseteq
N_{R}(p)$. If $m\in N_{R}(p)$, then, we get that $m\in N_{A}(p)$. Thus, there
exists $g=b_{0}+b_{1}Y_{1}+\cdots+b_{s}Y_{s}\in A$ such that

$\displaystyle
m=fg=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}X_{i}b_{j}Y_{j}\right)=\sum_{k=0}^{s+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(b_{j})X_{i}Y_{j}\right)$

Hence, we have $m=a_{0}b_{0}\in a_{0}R$, and thus $N_{R}(p)\subseteq a_{0}R$.
Therefore, we conclude that $N_{R}(p)=a_{0}R$ where $a_{0}\in N(R)$. ∎

##  4\. Nilpotent Associated Prime Ideals

It is well-known that associated prime ideals are an important tool in areas
such as commutative algebra and algebraic geometry concerning the primary
decomposition of ideals (e.g., Eisenbud [28, Chapter 3] and Eisenbud and
Harris [29, Section II.3.3]). Briefly, for a ring $R$ and a right $R$-module
$N_{R}$, the right annihilator of $N_{R}$ is denoted by $r_{R}(N_{R})$ =
$\left\\{r\in R\mid Nr=0\right\\}$, and $N_{R}$ is said to be prime if
$N_{R}\neq 0$ and $r_{R}(N_{R})=r_{R}(N_{R}^{{}^{\prime}})$, for every non-
zero submodule $N_{R}^{{}^{\prime}}\subseteq N_{R}$ [3, Definition 1.1]. If
$M_{R}$ is a right $R$-module, an ideal $P$ of $R$ is called an associated
prime of $M_{R}$ if there exists a prime submodule $N_{R}\subseteq M_{R}$ such
that $P=r_{R}(N_{R})$. The set of associated primes of $M_{R}$ is denoted by
${\rm Ass}(M_{R})$ [3, Definition 1.2]. One of the most important results on
these ideals was proved by Brewer and Heinzer [23] who showed that for a
commutative ring $R$, the associated primes ideals of the commutative
polynomial ring $R[x]$ are all extended, that is, every $P\in{\rm Ass}(R[x])$
may be expressed as $P=P_{0}[x]$, where $P_{0}=P\cap R\in{\rm Ass}(R)$ (Faith
[32] presented a proof of this fact using different algebraic techniques). In
several papers, Annin [1, 2, 3] extended the result above to the
noncommutative setting of Ore extensions, while Niño et al. [79] formulated
this result for skew PBW extensions.

Since Ouyang and Birkenmeier [82] introduced the notion of nilpotent
associated prime as a generalization of associated prime, and described all
nilpotent associated primes of the skew polynomial ring $R[x;\sigma,\delta]$
in terms of the nilpotent associated primes of $R$, a natural task is to
investigate this kind of ideals in the context of skew PBW extensions with the
aim of generalizing the results formulated in [82]. This is precisely the
purpose of this section.

We start by recalling some definitions presented by Ouyang and Birkenmeier
[82].

######  Definition 4.1.

Let $R$ be a ring.

  1. (i)

([82, Definition 3.1]) Let $I$ be a right ideal of a non-zero ring $R$. $I$ is
called a right quasi-prime ideal if $I\nsubseteq N(R)$ and
$N_{R}(I)=N_{R}(I^{\prime})$, for every right ideal $I^{\prime}\subseteq I$
and $I^{\prime}\nsubseteq N(R)$.

  2. (ii)

([82, Definition 3.2]) Let $N(R)$ be an ideal of a ring $R$. An ideal $P$ of
$R$ is called a nilpotent associated prime of $R$ if there exists a right
quasi-prime ideal $I$ such that $P=N_{R}(I)$. The set nilpotent associated
primes of $R$ is denoted by ${\rm NAss}(R)$.

An important concept in the characterization of nilpotent associated prime
ideals is the notion of good polynomial. These polynomials were first used by
Shock [100] with the aim of proving that uniform dimensions of a ring $R$ and
the polynomial ring $R[x]$ are equal (Annin [1, 2, 3] also considered such
polynomials in his study of associated prime ideals of Ore extensions). A
polynomial $f(x)\in R[x]$ is called good polynomial if it has the property
that the right annihilators of the coefficients of $f(x)$ are equal. Shock
proved that, given any non-zero polynomial $f(x)\in R[x]$, there exists $r\in
R$ such that $f(x)r$ is good [100, Proposition 2.2]. Ouyang and Birkenmeier
[82, Definition 3.3] introduced the notions of nilpotent degree and nilpotent
good polynomial to study the nilpotent associated prime ideals of Ore
extensions. Motivated by their ideas, we present the following definitions in
the setting of skew PBW extensions.

######  Definition 4.2.

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a skew PBW extension
over a ring $R$, and consider an element
$f=r_{0}+r_{1}X_{1}+\cdots+r_{k}X_{k}+\cdots+r_{l}X_{l}\notin N(R)A$, where
$X_{l}\succ X_{l-1}\succ\dots\succ X_{k}\succ X_{k-1}\succ\cdots\succ X_{1}$,
${\rm lm}(f)=x^{\alpha_{l}}$ and ${\rm lc}(f)=r_{l}$.

  1. (i)

If $r_{k}\notin N(R)$ and $r_{i}\in N(R)$, for all $i>k$, then we say that the
nilpotent degree of $f$ is $k$, which is denoted as ${\rm Ndeg}(f)$. If $f\in
N(R)A$, then we define ${\rm Ndeg}(f)=-1$.

  2. (ii)

Suppose that the nilpotent degree of $f$ is $k$. If $N_{R}(r_{k})\subseteq
N_{R}(r_{i})$, for all $i\leq k$, then we say that $f$ is a nilpotent good
polynomial.

The following result generalizes [82, Lemma 3.1] and [84, Lemma 2.16].

######  Theorem 4.3.

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a skew PBW extension
over a $(\Sigma,\Delta)$-compatible and NI ring $R$. For any
$f=r_{0}+r_{1}X_{1}+\cdots+r_{k}X_{k}+\cdots+r_{l}X_{l}\notin N(R)A$, there
exists $r\in R$ such that $fr$ is a nilpotent good polynomial.

###### Proof.

Let us assume that the result is false and suppose that
$f=r_{0}+r_{1}X_{1}+\dotsb+r_{k}X_{k}+\cdots+r_{l}X_{l}\notin N(A)$ is a
counterexample of minimal nilpotent degree ${\rm Ndeg}(f)=k$, that is, $fr$ is
not a nilpotent good polynomial. In particular, if $r=1$, we have that $f$ is
not a nilpotent good polynomial. Hence, there exists $i<k$ such that
$N_{R}(r_{k})\nsubseteq N_{r}(r_{i})$. Thus, we can find $b\in R$ such that
$b\in N_{R}(r_{k})$ and $b\notin N_{R}(r_{i})$, whence $r_{i}b\notin N(R)$ and
$r_{k}b\in N(R)$.

Notice that the degree $k$ coefficient of $fb$ is
$r_{k}\sigma^{\alpha_{k}}(b)+\sum_{i=k+1}^{l}r_{i}p_{\alpha_{i},b}$. Now,
$r_{k}\sigma^{\alpha_{k}}(b)\in N(R)$ due to the
$(\Sigma,\Delta)$-compatibility of $R$. On the other hand, we have ${\rm
Ndeg}(f)=k$, whence $r_{i}\in N(R)$, for all $i>k$. This means that
$r_{i}p_{\alpha_{i},b}\in N(R)$, for all $i>k$ and so
$\sum_{i=k+1}^{l}r_{i}p_{\alpha_{i},b}\in N(R)$ since $N(R)$ is an ideal of
$R$. Therefore, $fb$ has nilpotent degree at most $k-1$. Since $r_{i}b\notin
N(R)$, we have $fb\notin N(R)A$. By the minimality of $k$, there exists $c\in
R$ with $fbc$ a nilpotent good polynomial. However, this contradicts the fact
that $f$ is a counterexample, since $bc\in R$ and $f(bc)$ is a nilpotent good
polynomial. This concludes the proof. ∎

The following theorem characterizes the nilpotent associated primes ideals in
a skew PBW extension over a $(\Sigma,\Delta)$-compatible and NI ring. Later,
we state the different results from the literature that are corollaries of
Theorem 4.4.

######  Theorem 4.4.

If $A$ is a skew PBW extension over a $(\Sigma,\Delta)$-compatible and NI ring
$R$, then

${\rm NAss}(A)=\\{PA\mid P\in{\rm NAss}(R)\\}.$

###### Proof.

With the aim of establishing the desired equality, we proof the two
implications. Let $P\in{\rm NAss}(R)$. By Definition 4.1 (ii), there exists a
right ideal $I\nsubseteq N(R)$ with $I$ a right quasi-prime ideal of $R$ and
$P=N_{R}(I)$. Let us show that $PA=N_{A}(IA)$ and also that $IA$ is a quasi-
prime ideal. We show first that $PA=N_{A}(IA)$. Let
$i=a_{0}+a_{1}X_{1}+\cdots+a_{m}X_{m}\in IA$ and let
$f=b_{0}+b_{1}Y_{1}+\cdots+b_{l}Y_{l}\in PA$. Then,

$\displaystyle
if=\sum_{k=0}^{m+l}\left(\sum_{i+j=k}a_{i}X_{i}b_{j}Y_{j}\right)=\sum_{k=0}^{m+l}\left(\sum_{i+j=k}a_{i}\sigma^{\alpha_{i}}(b_{j})X_{i}Y_{j}+a_{i}p_{\alpha_{i},b_{j}}Y_{j}\right)$

Since $a_{i}b_{j}\in N(R)$ for every $i,j$, Proposition 2.11 implies that
$a_{i}\sigma^{\alpha_{i}}(b_{j})\in N(R)$, Furthermore, the polynomial
$p_{\alpha_{i},b_{j}}$ involves elements obtained evaluating $\sigma$’s and
$\delta$’s (depending on the coordinates of $\alpha_{i}$) in the element
$b_{j}$ by Proposition 2.5. This means that $a_{i}p_{\alpha_{i},b_{j}}\in
N(R)$ for every $i,j$ and thus, we obtain $if\in N(R)A=N(A)$. Hence,
$PA\subseteq N_{A}(IA)$.

Conversely, let $f=b_{0}+b_{1}Y_{1}+\cdots+b_{m}Y_{m}\in N_{A}(IA)$, then
$if\in N(A)=N(R)A$, for all $i=a_{0}+a_{1}X_{1}+\cdots+a_{l}X_{l}\in IA$,
whence $a_{i}\sigma^{\alpha_{i}}(b_{j})\in N(R)$, which implies that
$a_{i}b_{j}\in N(R)$, by Proposition 2.11. Since $a_{i}\in I$, we have
$b_{j}\in N_{R}(I)=P$. Therefore, we get $f\in PA$ and thus
$N_{A}(IA)\subseteq PA$. Hence, we conclude that $PA=N_{A}(IA)$.

Since the ideal $I$ is a right quasi-prime ideal, we have $I\nsubseteq N(R)$,
which implies that $IA\nsubseteq N(A)$. Let us show that for any right ideal
$U$ of $R$ it is satisfied that, if $U\nsubseteq N(A)$ and $U\subseteq IA$,
then $N_{A}(U)=N_{A}(IA)$. Let us see first that $N_{A}(IA)\subseteq
N_{A}(U)$. If $f=a_{0}+a_{1}X_{1}+\cdots+a_{m}X_{m}\in N_{A}(IA)$, this means
that $if\in N(A)$, for all $i\in IA$. In particular, since $U\subseteq IA$,
then $if\in N(A)$, for all $i\in U$, whence $f\in N_{A}(U)$. Hence, we have
$N_{A}(IA)\subseteq N_{A}(U)$.

Conversely, let $C_{U}\subseteq R$ consisting of all coefficients of elements
of $U$. Let us first consider $P^{\prime}$ the right ideal of $R$ generated by
$C_{U}$. Since $U\nsubseteq N(A)=N(R)A$, this means that $C_{U}\nsubseteq
N(R)$, and hence $P^{\prime}\subseteq I$ and $P^{\prime}\nsubseteq N(R)$.
Thus, as $I$ is a right quasi-prime ideal this implies that
$N_{R}(P^{\prime})=N_{R}(I)=P$. Now, if
$f=a_{0}+a_{1}X_{1}+\cdots+a_{m}X_{m}\in N_{A}(U)$ and
$u=u_{0}+u_{1}Y_{1}+\cdots+u_{t}Y_{t}\in U$, then $uf\in N(A)$, whence
$u_{i}\sigma^{\beta_{j}}(a_{j})\in N(R)$ and therefore $u_{i}a_{j}\in N(R)$,
for all $0\leq i\leq t$, $0\leq j\leq m$. Since $R$ is a NI ring, $N(R)$ is an
ideal, $u_{i}a_{j}\in N(R)$ implies $a_{j}u_{i}\in N(R)$ and thus
$u_{i}Ra_{j}\in N(R)$ gives that $(u_{i}Ra_{j})^{2}\in N(R)$. Therefore,

$a_{j}\in N_{R}(P^{\prime})=N_{R}(I)=P,\quad\text{for all}\quad 0\leq j\leq
m.$

Let $i=b_{0}+b_{1}Z_{1}+\cdots+b_{r}Z_{r}\in IA$. We have $b_{m}a_{j}\in
N(R)$, and so $b_{m}\sigma^{\alpha}(\delta^{\beta}(a_{j}))$ and
$a_{m}\delta^{\beta}(\sigma^{\alpha}(a_{j}))$ are elements of $N(R)$, for
every $\alpha,\beta\in\mathbb{N}^{n}$. Therefore, we get that $if\in
N(R)A=N(A)$, which implies that $f\in N_{A}(IA)$. Hence, we conclude
$N_{A}(U)\subseteq N_{A}(IA)$. Thus, we have proved that $PA=N_{A}(IA)$ and
also that $IA$ is a quasi-prime ideal.

Let $I\in{\rm NAss}(A)$. By Definition 4.1 (ii), there exists a right ideal
$J\nsubseteq N(A)$ with $J$ a right quasi-prime ideal of $A$ and $I=N_{A}(J)$.
Let $m=m_{0}+m_{1}X_{1}+\cdots+m_{k}X_{k}+\cdots+m_{n}X_{n}\notin N(A)=N(R)A$
and $m\in J$. Since $J\nsubseteq N(A)$, we may assume that $m$ is nilpotent
good and ${\rm Ndeg}(m)=k$, by Theorem 4.3. We consider $J_{0}=mA$ the
principal right ideal of $A$ generated by $m$. Since $m\notin N(A)=N(R)A$ this
implies that $J_{0}=mA\nsubseteq N(R)A=N(A)$, whence
$N_{A}(J)=N_{A}(J_{0})=N_{A}(mA)=I$ because $J$ is quasi-prime ideal. Now, we
consider the right ideal $m_{k}R$, and let us denote $U=N_{R}(m_{k}R)$.

Let us prove first that $I=UA$. Let $g=b_{0}+b_{1}Y_{1}+\cdots+b_{l}Y_{l}\in
UA$. Since $b_{j}\in U$, then $m_{k}Rb_{j}\in N(R)$ for all $0\leq j\leq l$.
Furthermore, $m$ is nilpotent good polynomial and ${\rm Ndeg}(m)=k$, hence
$m_{i}Rb_{j}\in N(R)$, for all $0\leq i\leq k$, and $0\leq j\leq l$. On the
other hand, for all $i>k$, $m_{i}\in N(R)$. Thus, we get $m_{i}Rb_{j}\in
N(R)$, for all $0\leq i\leq n$ and $0\leq j\leq l$. Now, for any element $h\in
A$ where $h=h_{0}+h_{1}Z_{1}+\cdots+h_{p}Z_{p}$, it is satisfied that
$m_{i}h_{d}b_{j}\in N(R)$, for all $0\leq i\leq n$, $0\leq d\leq p$ and $0\leq
j\leq l$. Therefore, by $(\Sigma,\Delta)$-compatibility of $R$, we obtain
$mhg\in N(A)$. This implies that $g\in N_{A}(mA)=I$, and so $UA\subseteq I$.
Conversely, let $g=b_{0}+b_{1}Y_{1}+\cdots+b_{l}Y_{l}\in I$. Since $mRg\in
N(A)$, it follows that $m_{i}Rb_{j}\in N(R)$, for all $0\leq i\leq n$, and
$0\leq j\leq l$. Thus, we have that $b_{j}\in N_{R}(m_{k}R)$, for all $0\leq
j\leq l$, and so $g\in UA$. Hence, we conclude $I\subseteq UA$ which implies
that $I=UA$.

Now, let us show that $m_{k}R$ is a quasi-prime ideal. Let $m_{k}R$ the
principal right ideal of $A$ generated by $m_{k}$. Since $m_{k}\notin N(R)$,
we have $m_{k}R\nsubseteq N(R)$. The idea is to show that for any right ideal
$Q\subseteq m_{k}R$ with $Q\nsubseteq N(R)$, it follows that
$N_{R}(Q)=N_{R}(m_{k}R)$. Assume that a right ideal $Q\subseteq m_{k}R$, and
$Q\nsubseteq N(R)$. Then $N_{R}(m_{k}R)\subseteq N_{R}(Q)$ by Proposition 3.3.
Now, we show that $N_{R}(Q)\subseteq N_{R}(m_{k}R)$. Let $W$ be the following
set $W=\left\\{mr\ |\ r\in Q\right\\}$, and let $WA$ the right ideal of $A$
generated by $W$.

First, notice that $WA\subseteq mA$. Since $Q\nsubseteq N(R)$, there exists
$a\in R$ such that $m_{k}a\in Q$ and $m_{k}a\notin N(R)$. If $m_{k}(m_{k}a)\in
N(R)$, then we have $m_{k}a\in N(R)$ which contradicts to the fact that
$m_{k}a\notin N(R)$. Thus $m_{k}(m_{k}a)\notin N(R)$, and hence
$m(m_{k}a)\notin N(A)$, by Proposition 2.17. This implies that $WA\nsubseteq
N(A)$. Since $J$ is quasi-prime ideal, we obtain $N_{A}(WA)=N_{A}(mA)=I$.

Suppose $q\in N_{R}(Q)$. Then $rq\in N(R)$, for each $r\in Q$. Now, for any
$mrf\in WA$ where $f=a_{0}+a_{1}Y_{1}+\cdots+a_{l}Y_{l}\in A$, the term of
$mrf$ is $m_{i}X_{i}ra_{j}Y_{j}$. The idea is to show that
$m_{i}X_{i}ra_{j}Y_{j}\in N(R)$. Since $rq\in N(R)$ and $N(R)$ is an ideal, it
follows that

$\displaystyle rq\in N(R)\Rightarrow qr\in N(R)\Rightarrow ra_{j}(qr)a_{j}q\in
N(R)\Rightarrow ra_{j}q\in N(R)$.

If $ra_{j}q\in N(R)$, then $m_{i}ra_{j}q\in N(R)$. Thus, due to the
$(\Sigma,\Delta)$-compatibility of $R$, we have that
$m_{i}X_{i}ra_{j}Y_{j}q\in N(R)A$ which implies that $mrfq\in N(R)A=N(A)$.
Hence, for any $\sum mr_{i}f_{i}\in WA$ it follows that $\sum(mr_{i}f_{i})q\in
N(A)$. Therefore, $q\in N_{A}(WA)=I=UA$, and so $q\in U=N_{R}(m_{k}R)$. Thus,
$N_{R}(Q)\subseteq N_{R}(m_{k}R)$, and this implies that
$N_{R}(Q)=N_{R}(m_{k}R)$. Hence, we conclude that $m_{k}R$ is quasi-prime
ideal. ∎

######  Corollary 4.5 ([82, Theorem 3.1]).

Let $R$ be a $(\sigma,\delta)$-compatible 2-primal ring. Then

${\rm NAss}(R[x;\sigma,\delta])=\\{P[x;\sigma,\delta]\mid P\in{\rm
NAss}(R)\\}.$

######  Corollary 4.6.

Let $R$ be a ring.

  1. (1)

([84, Theorem 3.1]) Let $R$ be a $\delta$-compatible reversible ring. Then

${\rm NAss}(R[x;\delta])=\\{P[x;\delta]\mid P\in{\rm NAss}(R)\\}.$

  2. (2)

([84, Corollary 3.2]) Let $R$ be a reversible ring. Then

${\rm NAss}(R[x])=\\{P[x]\mid P\in{\rm NAss}(R)\\}.$

######  Corollary 4.7.

Let $R$ be a ring and consider $M_{R}=R_{R}$.

  1. (1)

([79, Theorem 3.1.2]) Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a
skew PBW extension over $R$ and $M_{R}$ a right $R$-module. If $M_{R}$ is
$(\Sigma,\Delta)$-compatible, then

${\rm Ass}(M\langle X\rangle_{A})=\\{P\left\langle X\right\rangle\mid P\in{\rm
Ass}(R)\\}.$

  2. (2)

([3, Theorem 2.1]) Let $S=R[x;\sigma,\delta]$ and $M_{R}$ a right $R$-module.
If $M_{R}$ is $(\sigma,\delta)$-compatible, then

${\rm Ass}(M[X]_{S})=\\{P[X]\mid P\in{\rm Ass}(M_{R})\\}.$

  3. (3)

Let $S=R[x;\delta]$ be a differential polynomial ring of $R$ and $M_{R}$ a
right $R$-module. If $M_{R}$ is $\delta$-compatible, then

${\rm Ass}(M[X]_{S})=\\{P[X]\mid{P}\in{\rm Ass}(M_{R})\\}.$

######  Example 4.8.

Since Example 2.9 satisfies the conditions of Theorem 4.4, it follows that
${\rm NAss}(R[x;\overline{\sigma},\overline{\delta}])={\rm
NAss}(R)[x;\overline{\sigma},\overline{\delta}]$. On the other hand, the right
ideals of $R$ are given by

$I_{1}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}R,\ \ \ \ \ \ \ \ \
I_{2}=\begin{pmatrix}0&0\\\ 0&q(t)\end{pmatrix}R,\ \ \ \ \ \ \ \ \
I_{3}=\begin{pmatrix}0&p(t)\\\ 0&0\end{pmatrix}R$,

$I_{4}=\begin{pmatrix}0&p(t)\\\ 0&q(t)\end{pmatrix}R,\ \ \ \ \ \
I_{5}=\begin{pmatrix}q(t)&p(t)\\\ 0&0\end{pmatrix}R,\ \ \ \ \ \
I_{6}=\begin{pmatrix}q(t)&p(t)\\\ 0&s(t)\end{pmatrix}R$,

for $p(t),q(t)\in\Bbbk[t]$. Additionally, we can observe that the only quasi-
prime ideals of $R$ are $I_{2},I_{4},I_{5}$, and $I_{6}$. Furthermore,
$N_{R}(I_{2})=N_{R}(I_{4})=N_{R}(I_{5})=N_{R}(I_{6})=I_{3}$, where
$I_{3}=N(R)$. Thus, it follows that ${\rm
NAss}(R)=\left\\{N(R)\right\\}=\left\\{I_{3}\right\\}$. Therefore, we conclude
that ${\rm
NAss}(R[x;\overline{\sigma},\overline{\delta}])=N(R)[x;\overline{\sigma},\overline{\delta}]=I_{3}[x;\overline{\sigma},\overline{\delta}]$.

##  5\. Examples

As is clear, the relevance of the results presented in the paper is
appreciated when we apply them to algebraic structures more general than those
considered by Ouyang et al. [82, 84], i.e., noncommutative rings which cannot
be expressed as Ore extensions. With the aim of presenting different examples
of this kind of rings, in this section we consider several and remarkable
families of rings that have been studied in the literature which are
subfamilies of skew PBW extensions. Of course, the list of examples is not
exhaustive.

######  Definition 5.1 ([14]; [99, Definition C4.3]).

A $3$-dimensional algebra $\mathcal{A}$ is a $\Bbbk$-algebra generated by the
indeterminates $x,y,z$ subject to the relations $yz-\alpha zy=\lambda$,
$zx-\beta xz=\mu$, and $xy-\gamma yx=\nu$, where $\lambda,\mu,\nu\in\Bbbk
x+\Bbbk y+\Bbbk z+\Bbbk$, and $\alpha,\beta,\gamma\in\Bbbk^{*}$. $\mathcal{A}$
is called a 3-dimensional skew polynomial $\Bbbk$-algebra if the set
$\\{x^{i}y^{j}z^{k}\mid i,j,k\geq 0\\}$ forms a $\Bbbk$-basis of the algebra.

Different authors have studied ring-theoretical and geometrical properties of
3-dimensional skew polynomial algebras (e.g., [52, 57, 88, 89, 93]. Next, we
recall the classification of these algebras.

######  Proposition 5.2 ([99], Theorem C4.3.1]).

Up to isomorphism, a 3-dimensional skew polynomial $\Bbbk$-algebra
$\mathcal{A}$ is given by the following relations:

  1. (1)

If $|\\{\alpha,\beta,\gamma\\}|=3$, then $\mathcal{A}$ is defined by the
relations $yz-\alpha zy=0$, $zx-\beta xz=0$, and $xy-\gamma yx=0$.

  2. (2)

If $|\\{\alpha,\beta,\gamma\\}|=2$ and $\beta\not=\alpha=\gamma=1$, then
$\mathcal{A}$ is defined by one of the following rules:

    1. (a)

$yz-zy=z$, $zx-\beta xz=y$, and $xy-yx=x$.

    2. (b)

$yz-zy=z$, $zx-\beta xz=b$, and $xy-yx=x$.

    3. (c)

$yz-zy=0$, $zx-\beta xz=y$, and $xy-yx=0$.

    4. (d)

$yz-zy=0$, $zx-\beta xz=b$, and $xy-yx=0$.

    5. (e)

$yz-zy=az$, $zx-\beta xz=0$, and $xy-yx=x$.

    6. (f)

$yz-zy=z$, $zx-\beta xz=0$, and $xy-yx=0$.

Here $a,b\in\Bbbk$ are arbitrary; all non-zero values of $b$ yield isomorphic
algebras.

  3. (3)

If $\alpha=\beta=\gamma\not=1$, and if $\beta\not=\alpha=\gamma\not=1$, then
$\mathcal{A}$ is one of the following algebras:

    1. (a)

$yz-\alpha zy=0$, $zx-\beta xz=y+b$, and $xy-\alpha yx=0$.

    2. (b)

$yz-\alpha zy=0$, $zx-\beta xz=b$, and $xy-\alpha yx=0$.

Here $a,b\in\Bbbk$ is arbitrary; all non-zero values of $b$ yield isomorphic
algebras.

  4. (4)

If $\alpha=\beta=\gamma\not=1$, then $\mathcal{A}$ is determined by the
relations $yz-\alpha zy=a_{1}x+b_{1}$, $zx-\alpha xz=a_{2}y+b_{2}$, and
$xy-\alpha yx=a_{3}z+b_{3}$.

If $a_{i}=0$, then all non-zero values of $b_{i}$ yield isomorphic algebras.

  5. (5)

If $\alpha=\beta=\gamma=1$, then $\mathcal{A}$ is isomorphic to one of the
following algebras:

    1. (a)

$yz-zy=x$, $zx-xz=y$, and $xy-yx=z$.

    2. (b)

$yz-zy=0$, $zx-xz=0$, and $xy-yx=z$.

    3. (c)

$yz-zy=0$, $zx-xz=0$, and $xy-yx=b$.

    4. (d)

$yz-zy=-y$, $zx-xz=x+y$, and $xy-yx=0$.

    5. (e)

$yz-zy=az$, $zx-xz=x$, and $xy-yx=0$.

Here $a,b\in\Bbbk$ are arbitrary; all non-zero values of $b$ yield isomorphic
algebras.

It follows from Definition 5.1 that every $3$-dimensional skew polynomial
algebra is a skew PBW extension over the field $\Bbbk$, that is,
$\mathcal{A}\cong\sigma(\Bbbk)\left\langle x,y,z\right\rangle$. For these
algebras, Theorem 4.3 guarantees the existence of nilpotent good polynomials
and Theorem 4.4 characterizes the nilpotent associated prime ideals. In
particular, since $\Bbbk$ is a field (hence reduced), the only nilpotent
associated prime ideal is the zero ideal.

######  Example 5.3.

Following Havliček et al. [46, p. 79], the algebra
$U_{q}^{\prime}(\mathfrak{so}_{3})$ is generated over a field $\Bbbk$ by the
indeterminates $I_{1},I_{2}$, and $I_{3}$, subject to the relations given by

$I_{2}I_{1}-qI_{1}I_{2}=-q^{\frac{1}{2}}I_{3},\quad
I_{3}I_{1}-q^{-1}I_{1}I_{3}=q^{-\frac{1}{2}}I_{2},\quad
I_{3}I_{2}-qI_{2}I_{3}=-q^{\frac{1}{2}}I_{1},$

where $q\in\Bbbk\ \backslash\ \\{0,\pm 1\\}$. It is straightforward to show
that $U_{q}^{\prime}(\mathfrak{so}_{3})$ cannot be expressed as an iterated
Ore extension. However, this algebra is a skew PBW extension over $\Bbbk$,
i.e., $U_{q}^{\prime}(\mathfrak{so}_{3})\cong\sigma(\Bbbk)\langle
I_{1},I_{2},I_{3}\rangle$ [33, Example 1.3.3]. On the other hand, notice that
$\mathbb{R}$ is a $(\Sigma,\Delta)$-compatible NI ring with the identity
endomorphism of $\mathbb{R}$ and the trivial $\sigma$-derivation. Thus, since
that $r_{\Bbbk}(X)=N_{\Bbbk}(X)=\left\\{0\right\\}$, for all $X\nsubseteq
N(\Bbbk)$, Theorem 3.5 guarantees that
$N_{U_{q}^{\prime}(\mathfrak{so}_{3})}(U)$ is an ideal generated by a
nilpotent element of $U_{q}^{\prime}(\mathfrak{so}_{3})$, for each subset
$U\nsubseteq N(U_{q}^{\prime}(\mathfrak{so}_{3}))$. Following a similar
argument, by Theorem 3.10 $N_{U_{q}^{\prime}(\mathfrak{so}_{3})}(f\
U_{q}^{\prime}(\mathfrak{so}_{3}))$ is an ideal generated by a nilpotent
element, for each principal right ideal $f\
U_{q}^{\prime}(\mathfrak{so}_{3})\nsubseteq
N(U_{q}^{\prime}(\mathfrak{so}_{3}))$. Now, Theorem 3.15 guarantees that
$N_{U_{q}^{\prime}(\mathfrak{so}_{3})}(f)$ is an ideal generated by a
nilpotent element, for each $f\notin N(U_{q}^{\prime}(\mathfrak{so}_{3}))$.
Thinking about nilpotent good polynomials and nilpotent associated primes
ideals, Theorem 4.3 allows us to conclude that the algebra
$U_{q}^{\prime}(\mathfrak{so}_{3})$ has nilpotent good polynomials, while
Theorem 4.4 describes the nilpotent associated primes ideals for these
algebras.

######  Example 5.4.

Zhedanov [111, Section 1] introduced the Askey-Wilson algebra $AW(3)$ as the
algebra generated by three operators $K_{0},K_{1}$, and $K_{2}$, that satisfy
the commutation relations

(5.1) $\displaystyle[K_{0},K_{1}]_{\omega}$ $\displaystyle\ =K_{2},$ (5.2)
$\displaystyle[K_{2},K_{0}]_{\omega}$ $\displaystyle\
=BK_{0}+C_{1}K_{1}+D_{1},$ (5.3) $\displaystyle[K_{1},K_{2}]_{\omega}$
$\displaystyle\ =BK_{1}+C_{0}K_{0}+D_{0},$

where $B,C_{0},C_{1},D_{0},D_{1}$, are the structure constants of the algebra,
which Zhedanov assumes are real, and the $q$-commutator $[-,-]_{\omega}$ is
given by
$[\square,\triangle]_{\omega}:=e^{\omega}\square\triangle-e^{-\omega}\triangle\square$,
where $\omega$ is an arbitrary real parameter.

Notice that in the limit $\omega\to 0$, the algebra AW(3) becomes an ordinary
Lie algebra with three generators ($D_{0}$ and $D_{1}$ are included among the
structure constants of the algebra in order to take into account algebras of
Heisenberg-Weyl type).

The relations (5.1) - (5.3) can be written as

(5.4) $\displaystyle e^{\omega}K_{0}K_{1}-e^{-\omega}K_{1}K_{0}$
$\displaystyle\ =K_{2}$ (5.5) $\displaystyle
e^{\omega}K_{2}K_{0}-e^{-\omega}K_{0}K_{2}$ $\displaystyle\
=BK_{0}+C_{1}K_{1}+D_{1}$ (5.6) $\displaystyle
e^{\omega}K_{1}K_{2}-e^{-\omega}K_{2}K_{1}$ $\displaystyle\
=BK_{1}+C_{0}K_{0}+D_{0}.$

According to the relations that define the algebra, it is clear that AW(3)
cannot be expressed as an iterated Ore extension. Nevertheless, using
techniques such as those presented in [33, Theorem 1.3.1], it can be shown
that ${\rm AW(3)}$ is a skew PBW extension of endomorphism type, that is,
${\rm AW(3)}\cong\sigma(\mathbb{R})\langle K_{0},K_{1},K_{2}\rangle$. Since
that $\mathbb{R}$ is reduced,
$r_{\mathbb{R}}(X)=N_{\mathbb{R}}(X)=\left\\{0\right\\}$, for all $X\nsubseteq
N(\mathbb{R})$. Hence, the characterization the weak annihilators of $N_{{\rm
AW(3)}}(U)$, $N_{{\rm AW(3)}}(f\ {\rm AW(3)})$, and $N_{{\rm AW(3)}}(f)$, for
certain subsets, principal right ideals, and elements of algebra ${\rm
AW(3)}$, follow from Theorems 3.5, 3.10, and 3.15, respectively. Furthermore,
Theorems 3.8, 3.13 and 3.18 are also valid for the description of the weak
annihilators mentioned above. For AW(3), Theorem 4.3 guarantees the existence
of nilpotent good polynomials, and Theorem 4.4 describes the nilpotent
associated prime ideals. In particular, since that $\mathbb{R}$ is a field
(hence reduced), the only nilpotent associated prime ideal is the zero ideal.
All the theorems mentioned above are valid for ${\rm AW}(3)$ if we change
$\mathbb{R}$ to any reduced ring $R$.

######  Example 5.5.

As is well-known, algebras whose generators satisfy quadratic relations such
as Clifford algebras, Weyl-Heisenberg algebras, and Sklyanin algebras, play an
important role in analysis and mathematical physics. Motivated by these facts,
Golovashkin and Maximov [36] considered the algebras $Q(a,b,c)$, with two
generators $x$ and $y$, generated by the quadratic relations

(5.7) $yx=ax^{2}+bxy+cy^{2},$

where the coefficients $a,b$, and $c$ belong to an arbitrary field $\Bbbk$ of
characteristic zero. They studied conditions on these elements under which
such an algebra has a PBW basis of the form $\\{x^{m}y^{n}\mid
m,n\in\mathbb{N}\\}$.

In [36, Section 3], Golovashkin and Maximov presented a necessary and
sufficient condition for a PBW basis as above exists in the case $ac+b\neq 0$,
while in [36, Section 5], for the case $ac+b=0$, they proved that if $b\neq
0,-1$, then $Q(a,b,c)$ has a PBW basis, and if $b=-1$, then the elements
$\\{x^{m}y^{n}\mid m,n\in\mathbb{N}\\}$ are linearly independent but do not
form a PBW basis of $Q(a,b,c)$.

From relation (5.7) one can see that if $a,b,$ and $c$ are not zero
simultaneously, then $Q(a,b,c)$ is neither an Ore extension of $\Bbbk$,
$\Bbbk[x]$ nor of $\Bbbk[y]$. On the other hand, if $b\neq 0$ and $c=0$, then
one can check that $Q(a,b,c)$ is a skew PBW extension over $\Bbbk[x]$, i.e.,
$Q(a,b,c)\cong\sigma(\Bbbk[x])\langle y\rangle$. Note that $\Bbbk[x]$ is a
$(\Sigma,\Delta)$-compatible reduced ring with the identity endomorphism of
$\Bbbk[x]$ and the trivial $\sigma$-derivation. Since that $\Bbbk[x]$ is a
reduced ring, we get $r_{\Bbbk[x]}(X)=N_{\Bbbk[x]}(X)=\left\\{0\right\\}$, for
all $X\nsubseteq N(\Bbbk[x])$. Hence, the characterization the weak
annihilators of $N_{Q(a,b,c)}(U)$, $N_{Q(a,b,c)}(f\ Q(a,b,c))$, and
$N_{Q(a,b,c)}(f)$, for some subsets, principal right ideals, and elements of
algebra $Q(a,b,c)$, follow from Theorems 3.5, 3.10, and 3.15, respectively.
Also, Theorems 3.8, 3.13 and 3.18 are also valid for the description of these
weak annihilators. Lastly, for the algebra $Q(a,b,c)$, the existence of
nilpotent good polynomials and characterization of the nilpotent associated
prime ideals follow from Theorem 4.3 and 4.4, respectively. As an observation,
if $a=0$, then one can check that $Q(a,b,c)$ is a skew PBW extension on
$\Bbbk[y]$, i.e., $Q(a,b,c)\cong\sigma(\Bbbk[y])\langle x\rangle$ and all the
previous results are satisfied too. Similarly, if we change $\Bbbk$ to a
reduced ring $R$, all our observations and results are still valid for
$Q(a,b,c)$.

######  Example 5.6.

[25, Section 25.2] The basis of algebra
$\mathfrak{g}=\mathfrak{so}(5,\mathbb{C})$ consists of the elements
$\bf{J}_{\alpha\beta}=-{\bf J}_{\beta\alpha}$, $\alpha,\beta=1,2,3,4,5$
satisfying the commutation relations

(5.8)
$[\bf{J}_{\alpha\beta},\bf{J}_{\mu\nu}]=\delta_{\beta\mu}\bf{J}_{\alpha\nu}+\delta_{\alpha\nu}\bf{J}_{\beta\mu}-\delta_{\beta\nu}\bf{J}_{\alpha\mu}-\delta_{\alpha\mu}\bf{J}_{\beta\nu}.$

If we consider the elements

$\displaystyle{\bf H}_{1}$ $\displaystyle=\bf{iJ}_{12},$
$\displaystyle\bf{H}_{2}$ $\displaystyle=\bf{iJ}_{34},$
$\displaystyle\bf{E}_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}(\bf{J}_{45}+i\bf{J}_{35}),$
$\displaystyle\bf{E}_{2}$
$\displaystyle=\frac{1}{2}(\bf{J}_{23}+i\bf{J}_{13}-\bf{J}_{14}+i\bf{J}_{24}),$
$\displaystyle\bf{E}_{3}$
$\displaystyle=\frac{1}{\sqrt{2}}(\bf{J}_{15}-i\bf{J}_{25}),$
$\displaystyle\bf{E}_{4}$
$\displaystyle=\frac{1}{2}(\bf{J}_{23}+i\bf{J}_{13}+\bf{J}_{14}-i\bf{J}_{24}),$
$\displaystyle\bf{F}_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}(-\bf{J}_{45}+i\bf{J}_{35}),$
$\displaystyle\bf{F}_{2}$
$\displaystyle=\frac{1}{2}(-\bf{J}_{23}+i\bf{J}_{13}+\bf{J}_{14}+i\bf{J}_{24}),$
$\displaystyle\bf{F}_{3}$
$\displaystyle=\frac{1}{\sqrt{2}}(-\bf{J}_{15}-i\bf{J}_{25}),$
$\displaystyle\bf{F}_{4}$
$\displaystyle=\frac{1}{2}(-\bf{J}_{23}+i\bf{J}_{13}-\bf{J}_{14}-i\bf{J}_{24}),$

the commutation relations (5.8) can be expressed as

$\displaystyle[\bf{H}_{1},\bf{E}_{1}]$ $\displaystyle=0,$
$\displaystyle[\bf{H}_{1},\bf{E}_{2}]$ $\displaystyle=\bf{E}_{2},$
$\displaystyle[\bf{H}_{1},\bf{E}_{3}]$ $\displaystyle=\bf{E}_{3},$
$\displaystyle[\bf{H}_{1},\bf{E}_{4}]$ $\displaystyle=\bf{E}_{4},$
$\displaystyle[\bf{H}_{2},\bf{E}_{1}]$ $\displaystyle=\bf{E}_{1},$
$\displaystyle[\bf{H}_{2},\bf{E}_{2}]$ $\displaystyle=-\bf{E}_{2},$
$\displaystyle[\bf{H}_{2},\bf{E}_{3}]$ $\displaystyle=0,$
$\displaystyle[\bf{H}_{2},\bf{E}_{4}]$ $\displaystyle=\bf{E}_{4},$
$\displaystyle[\bf{H}_{1},\bf{F}_{1}]$ $\displaystyle=0,$
$\displaystyle[\bf{H}_{1},\bf{F}_{2}]$ $\displaystyle=-\bf{F}_{2},$
$\displaystyle[\bf{H}_{1},\bf{F}_{3}]$ $\displaystyle=-\bf{F}_{3},$
$\displaystyle[\bf{H}_{1},\bf{F}_{4}]$ $\displaystyle=-\bf{F}_{4},$
$\displaystyle[\bf{H}_{2},\bf{F}_{1}]$ $\displaystyle=-\bf{F}_{1},$
$\displaystyle[\bf{H}_{2},\bf{F}_{2}]$ $\displaystyle=\bf{F}_{2},$
$\displaystyle[\bf{H}_{2},\bf{F}_{3}]$ $\displaystyle=0,$
$\displaystyle[\bf{H}_{2},\bf{F}_{4}]$ $\displaystyle=-\bf{F}_{4},$
$\displaystyle[\bf{E}_{1},\bf{E}_{2}]$ $\displaystyle=\bf{E}_{3},$
$\displaystyle[\bf{E}_{1},\bf{E}_{3}]$ $\displaystyle=\bf{E}_{4},$
$\displaystyle[\bf{E}_{1},\bf{E}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{2},\bf{E}_{3}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{2},\bf{E}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{3},\bf{E}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{F}_{1},\bf{F}_{2}]$ $\displaystyle=-\bf{F}_{3}$
$\displaystyle[\bf{F}_{1},\bf{F}_{3}]$ $\displaystyle=-\bf{F}_{4},$
$\displaystyle[\bf{F}_{1},\bf{F}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{F}_{2},\bf{F}_{3}]$ $\displaystyle=0,$
$\displaystyle[\bf{F}_{2},\bf{F}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{F}_{3},\bf{F}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{1},\bf{F}_{1}]$ $\displaystyle=\bf{H}_{2},$
$\displaystyle[\bf{E}_{1},\bf{F}_{2}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{1},\bf{F}_{3}]$ $\displaystyle=-\bf{F}_{2},$
$\displaystyle[\bf{E}_{1},\bf{F}_{4}]$ $\displaystyle=-\bf{F}_{3},$
$\displaystyle[\bf{E}_{2},\bf{F}_{1}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{2},\bf{F}_{2}]$ $\displaystyle=\bf{H}_{1}-\bf{H}_{2},$
$\displaystyle[\bf{E}_{2},\bf{F}_{3}]$ $\displaystyle=\bf{F}_{1},$
$\displaystyle[\bf{E}_{2},\bf{F}_{4}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{3},\bf{F}_{1}]$ $\displaystyle=-\bf{E}_{2}$
$\displaystyle[\bf{E}_{3},\bf{F}_{2}]$ $\displaystyle=\bf{E}_{1},$
$\displaystyle[\bf{E}_{3},\bf{F}_{3}]$ $\displaystyle=\bf{H}_{1},$
$\displaystyle[\bf{E}_{3},\bf{F}_{4}]$ $\displaystyle=\bf{F}_{1},$
$\displaystyle[\bf{E}_{4},\bf{F}_{1}]$ $\displaystyle=-\bf{E}_{3}$
$\displaystyle[\bf{E}_{4},\bf{F}_{2}]$ $\displaystyle=0,$
$\displaystyle[\bf{E}_{4},\bf{F}_{3}]$ $\displaystyle=\bf{E}_{1},$
$\displaystyle[\bf{E}_{4},\bf{F}_{4}]$ $\displaystyle=\bf{H}_{1}+\bf{H}_{2}.$

Having in mind the classical PBW theorem for the universal enveloping algebra
$U(\mathfrak{so}(5,\mathbb{C}))$ of $\mathfrak{so}(5,\mathbb{C})$, and since
$U(\mathfrak{so}(5,\mathbb{C}))$ is a PBW extension of $\mathbb{C}$ [13,
Section 5], then $U(\mathfrak{so}(5,\mathbb{C}))$ is a skew PBW extension over
$\mathbb{C}$, i.e.,
$U(\mathfrak{so}(5,\mathbb{C}))\cong\sigma(\mathbb{C})\langle\bf{J}_{\alpha\beta}\mid
1\leq\alpha\leq\beta\leq 5\rangle$. The weak annihilator of certain subsets,
principal right ideals, and elements of algebra
$U(\mathfrak{so}(5,\mathbb{C}))$ are characterized by Theorems 3.5, 3.10, and
3.15, respectively. Furthermore, Theorems 3.8, 3.13 and 3.18 hold since
$U(\mathfrak{so}(5,\mathbb{C}))$ is a skew PBW extension of endomorphism type.
Theorem 4.3 asserts the existence of nilpotent good polynomials and Theorem
4.4 describes the nilpotent associated prime ideals of
$U(\mathfrak{so}(5,\mathbb{C}))$.

######  Example 5.7.

With the purpose of introducing generalizations of the classical bosonic and
fermionic algebras of quantum mechanics concerning several versions of the
Bose-Einstein and Fermi-Dirac statistics, Green [38] and Greenberg and Messiah
[39] introduced by means of generators and relations the parafermionic and
parabosonic algebras. For the completeness of the paper, briefly we recall the
definition of each one of these structures following the treatment developed
by Kanakoglou and Daskaloyannis [60]. Let
$[\square,\triangle]:=\square\triangle-\triangle\square$ and
$\\{\square,\triangle\\}:=\square\triangle+\triangle\square$.

Consider the $\Bbbk$-vector space $V_{F}$ freely generated by the elements
$f_{i}^{+},f_{j}^{-}$, with $i,j=1,\dotsc,n$. If $T(V_{F})$ is the tensor
algebra of $V_{F}$ and $I_{F}$ is the two-sided ideal $I_{F}$ generated by the
elements
$[[f_{i}^{\xi},f_{j}^{\eta}],f_{k}^{\varepsilon}]-\frac{1}{2}(\varepsilon-\eta)^{2}\delta_{jk}f_{i}^{\xi}+\frac{1}{2}(\varepsilon-\xi)^{2}\delta_{ik}f_{j}^{\eta}$,
for all values of $\xi,\eta,\varepsilon=\pm 1$, and $i,j,k=1,\dotsc,n$, then
the parafermionic algebra in $2n$ generators $P_{F}^{(n)}$ ($n$ parafermions)
is the quotient algebra of $T(V_{F})$ with the ideal $I_{F}$, that is,

$P_{F}^{(n)}=\frac{T(V_{F})}{\langle[[f_{i}^{\xi},f_{j}^{\eta}],f_{k}^{\varepsilon}]-\frac{1}{2}(\varepsilon-\eta)^{2}\delta_{jk}f_{i}^{\xi}+\frac{1}{2}(\varepsilon-\xi)^{2}\delta_{ik}f_{j}^{\eta}\mid\xi,\eta,\varepsilon=\pm
1,i,j,k=1,\dotsc,n\rangle}$

It is well-known (e.g., [60, Section 18.2]) that a parafermionic algebra
$P_{F}^{(n)}$ in $2n$ generators is isomorphic to the universal enveloping
algebra of the simple complex Lie algebra $\mathfrak{so}(2n+1)$ (according to
the classification of the simple complex Lie algebras, e.g., Kac [58, 59]),
i.e., $P_{F}^{(n)}\cong U(\mathfrak{so}(2n+1))$. On the other hand,
$P_{F}^{(n)}$ is a skew PBW extension over $\Bbbk$, that is,
$P_{F}^{(n)}\cong\sigma(\Bbbk)\langle f_{i}^{\xi},f_{j}^{\eta}\rangle$, for
values $\xi,\eta=\pm 1$, and $1\leq i,j\leq n$. Since $\Bbbk$ is a field
(hence reduced), then $r_{\Bbbk}(X)=N_{\Bbbk}(X)=\left\\{0\right\\}$, for all
$X\nsubseteq N(\Bbbk)$. This implies that the characterization the weak
annihilators of $N_{P_{F}^{(n)}}(U)$, $N_{P_{F}^{(n)}}(f\ P_{F}^{(n)})$, and
$N_{P_{F}^{(n)}}(f)$, for some subsets, principal right ideals, and elements
of algebra $P_{F}^{(n)}$, follow from Theorems 3.5, 3.10, and 3.15,
respectively. Furthermore, Theorems 3.8, 3.13 and 3.18 are also valid for the
description of these weak annihilators. We recall that these subsets,
principal right ideals and elements, are not contained in the set of nilpotent
elements of the algebra $P_{F}^{(n)}$. For the parafermionic algebra, Theorem
4.3 guarantees the existence of nilpotent good polynomials and Theorem 4.4
describes the nilpotent associated prime ideals.

Similarly, if $V_{B}$ denotes the $\Bbbk$-vector space freely generated by the
elements $b_{i}^{+},b_{j}^{-}$, $i,j=1,\dotsc,n$, $T(V_{B})$ is the tensor
algebra of $V_{B}$, and $I_{B}$ is the two-sided ideal of $T(V_{B})$ generated
by the elements
$[\\{b_{i}^{\xi},b_{j}^{\eta}\\},b_{k}^{\varepsilon}]-(\varepsilon-\eta)\delta_{jk}b_{i}^{\xi}-(\varepsilon-\xi)\delta_{ik}b_{j}^{\eta}$,
for all values of $\xi,\eta,\varepsilon=\pm 1$, and $i,j=1,\dotsc,n$, then the
parabosonic algebra $P_{B}^{(n)}$ in $2n$ generators ($n$ parabosons) is
defined as the quotient algebra $P_{B}^{(n)}/I_{B}$, that is,

$P_{B}^{(n)}=\frac{T(V_{B})}{\langle[\\{b_{i}^{\xi},b_{j}^{\eta}\\},b_{k}^{\varepsilon}]-(\varepsilon-\eta)\delta_{jk}b_{i}^{\xi}-(\varepsilon-\xi)\delta_{ik}b_{j}^{\eta}\mid\xi,\eta,\varepsilon=\pm
1,i,j=1,\dotsc,n\rangle}$

It is known that the parabosonic algebra $P_{B}^{(n)}$ in $2n$ generators is
isomorphic to the universal enveloping algebra of the classical simple complex
Lie superalgebra $B(0,n)$, that is, $P_{B}^{(n)}\cong U(B(0,n))$. For more
details about parafermionic and parabosonic algebras, see [60, Proposition
18.2] and references therein.

Similar to the case of parafermionic algebra, the parabosonic algebra is a
skew PBW extension over $\Bbbk$, that is,
$P_{B}^{(n)}\cong\sigma(\Bbbk)\langle b_{i}^{\xi},b_{j}^{\eta}\rangle$ for
values $\xi,\eta=\pm 1$, and $1\leq i,j\leq n$. The results mentioned for the
case of parafermionic algebras hold for parabasonic algebras. The weak
annihilator of some subsets, principal right ideals, and elements of algebra
$P_{B}^{(n)}$ are characterized by Theorems 3.5, 3.10, and 3.15, respectively.
Furthermore, Theorems 3.8, 3.13 and 3.18 are valid since $P_{B}^{(n)}$ is a
skew PBW extension of endomorphism type. Finally, thinking about the theory of
nilpotent associated prime ideals, Theorem 4.3 guarantees the existence of
nilpotent good polynomials, and Theorem 4.4 describes the nilpotent associated
prime ideals of $P_{B}^{(n)}$.

######  Example 5.8.

Other algebraic structures that illustrate the results obtained in the paper
concerns examples of generalized Weyl algebras, down-up algebras, and ambiskew
polynomial rings. For the completeness of the paper, we will briefly present
the definitions and some relations between these algebras (see [54, 55, 56,
57] for a detailed description).

Given an automorphism $\sigma$ and a central element $a$ of a ring $R$, Bavula
[9] defined the generalized Weyl algebra $R(\sigma,a)$ as the ring extension
of $R$ generated by the indeterminates $X^{-}$ and $X^{+}$ subject to the
relations

$X^{-}X^{+}=a,\qquad X^{+}X^{-}=\sigma(a),$

and, for all $b\in R$,

$X^{+}b=\sigma(b)X^{+},\qquad X^{-}\sigma(b)=bX^{-}.$

This family of algebras includes the classical Weyl algebras, primitive
quotients of $U(\mathfrak{sl}_{2})$, and ambiskew polynomial rings (see below
the definition of these objects). Generalized Weyl algebras have been
extensively studied in the literature by various authors (see [9, 10, 11, 56],
and references therein).

The down-up algebras $A(\alpha,\beta,\gamma)$, where
$\alpha,\beta,\gamma\in\mathbb{C}$, were defined by Benkart and Roby [15, 16]
as generalizations of algebras generated by a pair of operators, precisely,
the “down” and “up” operators, acting on the vector space $\mathbb{C}P$ for
certain partially ordered sets $P$. Let us see the details.

Consider a partially ordered set $P$ and let $\mathbb{C}P$ be the complex
vector space with basis $P$. If for an element $p$ of $P$, the sets $\\{x\in
P\mid x\succ p\\}$ and $\\{x\in P\mid x\prec p\\}$ are finite, then we can
define the “down” operator $d$ and the “down” operator $u$ in ${\rm
End}_{\mathbb{C}}\mathbb{C}P$ as $u(p)=\sum_{x\succ p}x$ and
$d(p)=\sum_{x\prec p}x$, respectively (for partially ordered sets in general,
one needs to complete $\mathbb{C}P$ to define $d$ and $u$). For any
$\alpha,\beta,\gamma\in\mathbb{C}$, the down-up algebra is the
$\mathbb{C}$-algebra generated by $d$ and $u$ subject to the relations

(5.9) $\displaystyle d^{2}u=\alpha dud+\beta ud^{2}+\gamma d,$ (5.10)
$\displaystyle du^{2}=\alpha udu+\beta u^{2}d+\gamma u.$

A partially ordered set $P$ is called $(q,r)$-differential if there exist
$q,r\in\mathbb{C}$ such that the down and up operators for $P$ satisfy the
relations (5.9) and (5.10), and $\alpha=q(q+1),\beta=-q^{3}$, and $\gamma=r$.
From [16], we know that for $0\neq\lambda\in\mathbb{C}$,
$A(\alpha,\beta,\gamma)\simeq A(\alpha,\beta,\lambda\gamma)$. This means that
when $\gamma\neq 0$, no problem if we assume $\gamma=1$. For more details
about the combinatorial origins of down-up algebras, see [15, Section 1].

Remarkable examples of down-up algebras include the enveloping algebra of the
Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ and some of its deformations
introduced by Witten [107] and Woronowicz [109]. Related to the theoretical
properties of these algebras, Kirkman et al. [63] proved that a down-up
algebra $A(\alpha,\beta,\gamma)$ is Noetherian if and only if $\beta$ is non-
zero. As a matter of fact, they showed that $A(\alpha,\beta,\gamma)$ is a
generalized Weyl algebra in the sense of Bavula [9], and that
$A(\alpha,\beta,\gamma)$ has a filtration for which the associated graded ring
is an iterated Ore extension over $\mathbb{C}$.

Following [15, p. 32], if $\mathfrak{g}$ is a 3-dimensional Lie algebra over
$\mathbb{C}$ with basis $x,y,[x,y]$ such that $[x,[x,y]]=\gamma x$ and
$[[x,y],y]=\gamma y$, then in the universal enveloping algebra
$U(\mathfrak{g})$ of $\mathfrak{g}$ these relations are given by

$\displaystyle x^{2}y-2xyx+yx^{2}=$ $\displaystyle\ \gamma x,$ $\displaystyle
xy^{2}-2yxy+y^{2}x=$ $\displaystyle\ \gamma y.$

Notice that $U(\mathfrak{g})$ is a homomorphic algebra of the down-up algebra
$A(2,-1,\gamma)$ via the mapping $\phi:A(2,-1,\gamma)\to U(\mathfrak{g})$,
$d\mapsto x,u\mapsto y$, and the mapping $\psi:\mathfrak{g}\to
A(2,-1,\gamma)$, $x\mapsto d,y\mapsto u$, $[x,y]\mapsto du-ud$, extends by the
universal property of $U(\mathfrak{g})$ to an algebra homomorphism
$\psi:U(\mathfrak{g})\to A(2,-1,\gamma)$ which is the inverse of $\psi$, then
$U(\mathfrak{g})$ is isomorphic to $A(2,-1,\gamma)$.

The well-known Lie algebra $\mathfrak{sl}_{2}$ of $2\times 2$ complex matrices
of trace zero has a standard basis $e,f,h$, which satisfies $[e,f]=h,\
[h,e]=2e$, and $[h,f]=-2f$. It is straightforward to see that
$U(\mathfrak{sl}_{2})\cong A(2,-1,-2)$. Also, for the Heisenberg Lie algebra
$\mathfrak{h}$ with basis $x,y,z$ where $[x,y]=z$ and $[z,\mathfrak{h}]=0$,
$U(\mathfrak{h})\cong A(2,-1,0)$.

With the aim of providing an explanation of the existence of quantum groups,
Witten [107, 108] introduced a 7-parameter deformation of the universal
enveloping algebra $U(\mathfrak{sl}_{2})$. By definition, Witten’s deformation
is a unital associative algebra over a field $\Bbbk$ (which is algebraically
closed of characteristic zero) that depends on a $7$-tuple
$\underline{\xi}=(\xi_{1},\dotsc,\xi_{7})$ of elements of $\Bbbk$. This
algebra is generated by the indeterminates $x,y,z$ subject to the defining
relations

(5.11) $\displaystyle xz-\xi_{1}zx=$ $\displaystyle\ \xi_{2}x,$ (5.12)
$\displaystyle zy-\xi_{3}yz=$ $\displaystyle\ \xi_{4}y,$ (5.13) $\displaystyle
yx-\xi_{5}xy=$ $\displaystyle\ \xi_{6}z^{2}+\xi_{7}z,$

and is denoted by $W(\underline{\xi})$. By [15, Section 2], if $\xi_{6}=0$ and
$\xi_{7}\neq 0$ then it is easy to see that expressions (5.11), (5.12), and
(5.13) imply the equalities

$\displaystyle-\xi_{5}x^{2}y+(1+\xi_{1}\xi_{5})xyx-\xi_{1}yx^{2}=$
$\displaystyle\ \xi_{2}\xi_{7}x,$
$\displaystyle-\xi_{5}xy^{2}+(1+\xi_{3}\xi_{5})yxy-\xi_{3}y^{2}x=$
$\displaystyle\ \xi_{4}\xi_{7}y.$

When $\xi_{5}\neq 0,\xi_{1}=\xi_{3}$, and $\xi_{2}=\xi_{4}$, we obtain

$\displaystyle x^{2}y=$ $\displaystyle\
\frac{1+\xi_{1}\xi_{5}}{\xi_{5}}xyx-\frac{\xi_{1}}{\xi_{5}}yx^{2}-\frac{\xi_{2}\xi_{7}}{\xi_{5}}x,$
$\displaystyle xy^{2}=$ $\displaystyle\
\frac{1+\xi_{1}\xi_{5}}{\xi_{5}}yxy-\frac{\xi_{1}}{\xi_{5}}y^{2}x-\frac{\xi_{2}\xi_{7}}{\xi_{5}}y.$

As one can check, a Witten deformation algebra $W(\underline{\xi})$ with
$\xi_{6}=0,\xi_{5}\xi_{7}\neq 0,\xi_{1}=\xi_{3}$, and $\xi_{2}=\xi_{4}$ is a
homomorphic image of the down-up algebra $A(\alpha,\beta,\gamma)$ where

(5.14)
$\alpha=\frac{1+\xi_{1}\xi_{5}}{\xi_{5}},\quad\beta=-\frac{\xi_{1}}{\xi_{5}},\quad\gamma=-\frac{\xi_{2}\xi_{7}}{\xi_{5}}.$

All these facts allow to prove that a Witten deformation algebra $W(\xi)$ with

(5.15) $\xi_{6}=0,\quad\xi_{5}\xi_{7}\neq 0,\quad\xi_{1}=\xi_{3},\quad{\rm
and}\quad\xi_{2}=\xi_{4},$

is isomorphic to the down-up algebra $A(\alpha,\beta,\gamma)$ with
$\alpha,\beta,\gamma$ given by (5.14). Notice that any down-up algebra
$A(\alpha,\beta,\gamma)$ with not both $\alpha$ and $\beta$ equal to $0$ is
isomorphic to a Witten deformation algebra $W(\underline{\xi})$ whose
parameters satisfy (5.15).

Since algebras $W(\underline{\xi})$ are filtered, Le Bruyn [66, 67] studied
the algebras $W(\underline{\xi})$ whose associated graded algebras are
Auslander regular. He determined a 3-parameter family of deformation algebras
which are said to be conformal $\mathfrak{sl}_{2}$ algebras that are generated
by the indeterminates $x,y,z$ over a field $\Bbbk$ subject to the relations
given by

(5.16) $zx-axz=x,\quad zy-ayz=y,\quad yx-cxy=bz^{2}+z.$

In the case $c\neq 0$ and $b=0$, the conformal $\mathfrak{sl}_{2}$ algebra
with defining relations given by (5.16) is isomorphic to the down-up algebra
$A(\alpha,\beta,\gamma)$ with $\alpha=c^{-1}(1+ac),\beta=-ac^{-1}$ and
$\gamma=-c^{-1}$. Notice that if $c=b=0$ and $a\neq 0$, then the conformal
$\mathfrak{sl}_{2}$ algebra is isomorphic to the down-up algebra
$A(\alpha,\beta,\gamma)$ with $\alpha=a^{-1},\beta=0$, and $\gamma=-a^{-1}$.
As one can check using (5.16), conformal $\mathfrak{sl}_{2}$ algebras are not
Ore extensions but skew PBW extensions over $\Bbbk[z]$.

Of interest for the examples in the paper, Kulkarni [65] showed that under
certan assumptions on the parameters, a Witten deformation algebra is
isomorphic to a conformal $\mathfrak{sl}_{2}$ algebra or to an iterated Ore
extension (double Ore extension). More exactly, following [65, Theorem 3.0.3]
if $\xi_{1}\xi_{3}\xi_{5}\xi_{2}\neq 0$ or $\xi_{1}\xi_{3}\xi_{5}\xi_{4}\neq
0$, then $W(\underline{\xi})$ is isomorphic to one of the following algebras:

  1. (1)

A conformal $\mathfrak{sl}_{2}$ algebra with generators $x,y,z$ and relations
given by (5.16), for some elements $a,b,c\in\Bbbk$.

  2. (2)

An iterated Ore extension whose generators satisfy

    1. (i)

$xz-zx=x$, $zy-yz=\zeta y$, $yx-\eta xy=0$, or

    2. (ii)

$xw=\theta wx$, $wy=\kappa yw$, $yx=\lambda xy$, for parameters
$\zeta,\eta,\theta,\kappa,\lambda\in\Bbbk$.

Considering the above facts and theorems presented in this paper, we can
assert that weak annihilators of subsets, principal right ideals, and elements
of conformal $\mathfrak{sl}_{2}$ algebras are characterized by Theorems 3.5,
3.10, and 3.15, respectively. About the theory of nilpotent associated prime
ideals, Theorem 4.3 establishes the existence of nilpotent good polynomials,
and Theorem 4.4 describes the nilpotent associated prime ideals of the
algebra.

Jordan [56] introduced a certain class of iterated Ore extensions
$R(B,\sigma,c,p)$ called ambiskew polynomial rings (these structures have been
studied by Jordan at various levels of generality in several papers [52, 53,
54, 55]) that contains different examples of noncommutative algebras. We
recall the treatment of ambiskew polynomial rings in the framework appropriate
to down-up algebras with $\beta\neq 0$ following [56, 57].

For $\Bbbk$ a field and $\Bbbk^{*}=\Bbbk\ \backslash\ \\{0\\}$ its
multiplicative group, consider a commutative $\Bbbk$-algebra $B$, a
$\Bbbk$-automorphism of $B$, and elements $c\in B$ and $p\in\Bbbk^{*}$. Let
$S$ be the Ore extension $B[x;\sigma^{-1}]$ and extend $\sigma$ to $S$ by
setting $\sigma(x)=px$. By [26, p. 41], there is a $\sigma$-derivation
$\delta$ of $S$ such that $\delta(B)=0$ and $\delta(x)=c$. The ambiskew
polynomial ring $R=R(B,\sigma,c,p)$ is the Ore extension $S[y;\sigma,\delta]$,
whence the following relations hold:

(5.17) $yx-pxy=c,\quad{\rm and,\ for\ all}\ b\in B,\quad
xb=\sigma^{-1}(b)x\quad{\rm and}\quad yb=\sigma(b)y.$

Equivalently, $R$ can be presented as
$R=B[y;\sigma][x;\sigma^{-1},\delta^{\prime}]$ with $\sigma(y)=p^{-1}y$,
$\delta^{\prime}(B)=0$, and $\delta^{\prime}(y)=-p^{-1}c$, so that
$xy-p^{-1}yx=-p^{-1}c$. If we consider the relation $xb=\sigma^{-1}(b)x$ as
$bx=x\sigma(b)$, then we can see that the definition involves twists from both
sides using $\sigma$; this is the reason for the name of the objects. As a
matter of fact, every generalized Weyl algebra is isomorphic to a factor of an
ambiskew polynomial ring [52], and the ambiskew polynomial ring
$A(R,\sigma,c,p)$ is isomorphic to the generalized Weyl algebra
$R[w](\sigma,w)$, where $w$ is extended to $R[w]$ by setting
$\sigma(w)=pw+\sigma(c)$. In the conformal case, $A(R,\sigma,c,p)\simeq
R[z](\sigma,z+\sigma(a))$, where $\sigma(z)=pz$. From the definition, it
follows that ambiskew polynomial rings are skew PBW extensions over $B$, that
is, $R(B,\sigma,c,p)\cong\sigma(B)\langle y,x\rangle$.

######  Example 5.9.

Recently, Bavula [11] defined the skew bi-quadratic algebras with the aim of
giving an explicit description of bi-quadratic algebras on 3 generators with
PBW basis. Let us recall the details of its definition.

For a ring $R$ and a natural number $n\geq 2$, a family $M=(m_{ij})_{i>j}$ of
elements $m_{ij}\in R$ ($1\leq j<i\leq n$) is called a lower triangular half-
matrix with coefficients in $R$. The set of all such matrices is denoted by
$L_{n}(R)$.

If $Z(R)$ denotes the center of $R$, $\sigma=(\sigma_{1},\dotsc,\sigma_{n})$
is an $n$-tuple of commuting endomorphisms of $R$,
$\delta=(\delta_{1},\dotsc,\delta_{n})$ is an $n$-tuple of
$\sigma$-endomorphisms of $R$ (that is, $\delta_{i}$ is a
$\sigma_{i}$-derivation of $R$ for $i=1,\dotsc,n$), $Q=(q_{ij})\in
L_{n}(Z(R))$, $\mathbb{A}:=(a_{ij,k})$ where $a_{ij,k}\in R$, $1\leq j<i\leq
n$ and $k=1,\dotsc,n$, and $\mathbb{B}:=(b_{ij})\in L_{n}(R)$, the skew bi-
quadratic algebra) (SBQA)
$A=R[x_{1},\dotsc,x_{n};\sigma,\delta,Q,\mathbb{A},\mathbb{B}]$ is a ring
generated by the ring $R$ and elements $x_{1},\dotsc,x_{n}$ subject to the
defining relations

(5.18) $\displaystyle x_{i}r=$ $\displaystyle\
\sigma_{i}(r)x_{i}+\delta_{i}(r),\quad{\rm for}\ i=1,\dotsc,n,\ {\rm and\
every}\ r\in R,$ (5.19) $\displaystyle x_{i}x_{j}-q_{ij}x_{j}x_{i}=$
$\displaystyle\ \sum_{k=1}^{n}a_{ij,k}x_{k}+b_{ij},\quad{\rm for\ all}\ j<i.$

In the particular case, when $\sigma_{i}={\rm id}_{R}$ and $\delta_{i}=0$, for
$i=1,\dotsc,n$, the ring $A$ is called the bi-quadratic algebra (BQA) and is
denoted by $A=R[x_{1},\dotsc,x_{n};Q,\mathbb{A},\mathbb{B}]$. $A$ has PBW
basis if $A=\bigoplus_{\alpha\in\mathbb{N}^{n}}Rx^{\alpha}$ where
$x^{\alpha}=x_{1}^{\alpha_{1}}\dotsb x_{n}^{\alpha_{n}}$.

It is clear from the definition that
$R[x_{1},\dotsc,x_{n};\sigma,\delta,Q,\mathbb{A},\mathbb{B}]\cong\sigma(R)\langle
x_{1},\dotsc,x_{n}\rangle$, and, as a matter of fact, skew PBW extensions are
more general than skew bi-quadratic algebras since the former do not assume
the commutativity of $\sigma$’s (c.f. [70, Definition 2.1]) and the conditions
about the membership of the elements $q_{ij}$ and $b_{ij}$.

Notice that if $R$ is a $(\Sigma,\Delta)$-compatible NI ring, then the
characterization of the weak annihilators $N_{A}(U)$, $N_{A}(fA)$, and
$N_{A}(f)$, for subsets, principal right ideals, and elements of algebra
$A=R[x_{1},\dotsc,x_{n};Q,\mathbb{A},\mathbb{B}]$, respectively, follow from
Theorems 3.5, 3.10, and 3.15. If, in addition, $R$ is a field, Theorems 3.8,
3.13 and 3.18 are also valid for the description of all of them. Finally, for
the bi-quadratic algebras, if $R$ is a $(\Sigma,\Delta)$-compatible NI ring,
then Theorem 4.3 guarantees the existence of nilpotent good polynomials, and
Theorem 4.4 describes its nilpotent associated prime ideals.

##  6\. Future work

In this paper, we have studied weak annihilators and nilpotent associated
primes of skew PBW extensions over $(\Sigma,\Delta)$-compatible rings, where
some results were shown in the more general setting of weak compatibility.
Having in mind that Louzari, Niño, Ramírez and Reyes [74, 79, 91] considered
the notion of compatibility (and relater ring-theoretical properties) for
modules over these extensions, a first and natural task is to characterize
nilpotent associated prime ideals over these objects.

Related with this, since Macdonald [78] introduced a dual theory to primary
decomposition called secondary representation whose key objects are known as
attached primes, which were considered by Annin [4, 5] with the aim of
studying the behavior of the attached prime ideals of inverse polynomial
modules over skew polynomial rings of automorphism type $R[x;\sigma]$, a
second task is to determine the attached primes of polynomial modules over
skew PBW extensions with the aim of generalizing Annin’s results and others
related in the Module and Ring Theory.

## References

  * [1] S. Annin, Associated and Attached Primes Over Noncommutative Rings, PhD Thesis. University of California, Berkeley, (2002).
  * [2] S. Annin, Associated primes over skew polynomial ring, Comm. Algebra 30 (5) (2002) 2511–2528.
  * [3] S. Annin, Associated primes over Ore extension rings, J. Algebra Appl. 3 (2) (2004) 193–205.
  * [4] S. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra 212 (3) (2008) 510–521.
  * [5] S. Annin, Attached primes under skew polynomial extensions, J. Algebra Appl. 10 (3) (2011) 537–547.
  * [6] J. Apel, Gröbnerbasen in nichtkommutativen Algebren und ihre Anwendung, Ph.D. Thesis, University of Leipzig (1988).
  * [7] E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974) 470–473.
  * [8] V. A. Artamonov, Derivations of skew PBW extensions, Commun. Math. Stat. 3 (4) (2015) 449–457.
  * [9] V. V. Bavula, Generalized Weyl algebras and their representations, St. Petersburg Math. J. 4 (1) (1992) 71–92.
  * [10] V. V. Bavula, Global dimension of Generalized Weyl algebras, Canad. Math. Soc. Confe. Proce. 18 (1996) 81–107.
  * [11] V. Bavula, Description of bi-quadratic algebras on 3 generators with PBW basis, (2021) https://arxiv.org/abs/2112.11105v1
  * [12] J. A. Beachy, W. D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1) (1975) 1–13.
  * [13] A. D. Bell, K. R. Goodearl, Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions, Pacific J. Math. 131 (1) (1988) 13–37.
  * [14] A. D. Bell, S. P. Smith, Some 3-dimensional skew polynomial rings. University of Wisconsin, Milwaukee, preprint, (1990).
  * [15] G. Benkart, Down-up algebras and Witten’s deformation of the universal enveloping algebra of $\mathfrak{sl}_{2}$, in: S. Geun Hahn, H. Chul Myung, E. Zelmanov (Eds.), Recent Progress in Algebra, Contemporary Mathematics, Vol. 224, Amer. Math. Soc., Providence, 1998, pp. 29–45.
  * [16] G. Benkart, T. Roby, Down-up algebras, J. Algebra 209 (1) (1998) 305–344.
  * [17] S. K. Berberian, Baer $*$-Rings, Springer Science and Business Media, 2010.
  * [18] V. K. Bhat, Associated Prime Ideals of Skew Polynomial Rings, Beitr Algebra Geom. 49 (1) (2008) 277–283.
  * [19] V. K. Bhat, Associated prime ideals of weak $\sigma$-rigid rings and their extensions, Algebra Discrete Math. 10 (1) (2010) 8–17.
  * [20] G. F. Birkenmeier, G. F., J. Y. Kim, J. K. Park, Principally quasi-Baer rings. Comm. Algebra 29(2) (2001) 639–660.
  * [21] G. F. Birkenmeier, H. E. Heatherly, E. K. Lee, Completely prime ideals and associated radicals, in: S. K. Jain, S. T. Rizvi (Eds.), Ring Theory. Granville, OH, 1992. World Scientific, Singapore and River Edge (1993) 102–129.
  * [22] G. F. Birkenmeier, J. K. Park, S. T. Rizvi, Extensions of Rings and Modules, Birkhäuser, New York 2013.
  * [23] J. Brewer, W. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1) (1974) 1–7.
  * [24] J. Bueso, J. Gómez-Torrecillas, A. Verschoren, Algorithmic Methods in Non-commutative Algebra: Applications to Quantum Groups, Dordrecht, Kluwer, 2003.
  * [25] Č. Burdík, O. Navrátil, Decomposition of the Enveloping Algebra so(5), in: S. Silvestrov, E. Paal, V. Abramov, A. Stolin (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer, Berlin, Heidelberg, 2009, pp. 297–302.
  * [26] P. M. Cohn, Free Rings and Their Relations, Second Edition. Academic Press, London, 1985.
  * [27] P. M. Cohn, Reversible rings, Bull. Lond. Math. Soc. 31 (6) (1999) 641–648.
  * [28] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, Vol. 150, Springer-Verlag, 1995.
  * [29] D. Eisenbud, J. Harris, The Geometry of Schemes, Graduate Texts in Mathematics, Vol. 197, Springer-Verlag, 2000.
  * [30] C. Faith, Rings with zero intersection property on annihilators: ZIP rings, Publ. Math. 33 (2) (1989) 329–338.
  * [31] C. Faith, Annihilator ideals, associated primes and Kasch-McCoy commutative rings, Comm. Algebra 19 (7) (1991) 1867–1892.
  * [32] C. Faith, Associated primes in commutative polynomial rings, Comm. Algebra 28 (8) (2000) 3983–3986.
  * [33] W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez, H. Venegas, Skew PBW Extensions: Ring and Module-theoretic properties, Matrix and Gröbner Methods, and Applications, Springer, Cham, 2020.
  * [34] P. Gabriel, Représentations Des algébres de Lie résolubles, Sem. Bourbaki 347 1–22. Lecture Notes in Math. 179. Springer-Verlag. Berlin (1968–69).
  * [35] C. Gallego, O. Lezama, Gröbner bases for ideals of $\sigma$-PBW extensions, Comm. Algebra 39 (1) (2011) 50–75.
  * [36] A. V. Golovashkin, V. M. Maximov, On algebras of skew polynomials generated by quadratic homogeneous relations, J. Math. Sci. (N. Y.) 129 (2) (2005) 3757 – 3771.
  * [37] J. G. Gómez Torrecillas, Basic Module Theory over Non-commutative Rings with Computational Aspects of Operator Algebras, in: M. Barkatou, T. Cluzeau, G. Regensburger, M. Rosenkranz (Eds.) Algebraic and Algorithmic Aspects of Differential and Integral Operators, AADIOS 2012, Lecture Notes in Computer Science, Vol. 8372, Springer, Berlin, Heidelberg, 2014, pp. 23–82.
  * [38] H. S. Green, A generalized method of field quantization, Phys. Rev. 90 (2) (1953) 270.
  * [39] O. W. Greenberg, A. M. L. Messiah, Selection rules for parafields and the absence of paraparticles in nature, Phys. Rev. 138 (5B) (1965) B1155–B1167.
  * [40] M. Hamidizadeh, E. Hashemi, A. Reyes, A classification of ring elements in skew PBW extensions over compatible rings, Int. Electron. J. Algebra 28 (1) (2020) 75–97.
  * [41] E. Hashemi, Kh. Khalilnezhad, A. Alhevaz, $(\Sigma,\Delta)$-Compatible Skew PBW Extension Ring, Kyungpook Math. J. 57 (3) (2017) 401–417.
  * [42] E. Hashemi, Kh. Khalilnezhad, A. Alhevaz, Extensions of rings over 2-primal rings, Matematiche 74 (1) (2019) 141–162.
  * [43] E. Hashemi, Kh. Khalilnezhad, M. Ghadiri, Baer and quasi-baer properties of skew PBW extensions, J. Algebr. Syst. 7 (1) (2019) 1–24.
  * [44] E. Hashemi, A. Moussavi, H. Haj Seyyed Javadi, Polynomial Ore extensions of Baer and p.p.-rings, Bull. Iranian Math. Soc. 29 (2) (2003) 65–86.
  * [45] E. Hashemi, A. Moussavi, Polinomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (3) (2005) 207–224.
  * [46] M. Havlíček, A. U. Klimyk, S. Pošta, Central elements of the algebras $U^{\prime}(\mathfrak{so}_{m})$ and $U(\mathfrak{iso}_{m})$, Czech. J. Phys. 50 (1) (2000) 79–84.
  * [47] C. Y. Hong, T. K. Kwak, S. T. Rizvi, Extensions of generalized Armendariz rings, Algebra Colloq. 13 (2) (2006) 253–266.
  * [48] S. Higuera, A. Reyes, A survey on the fusible property of skew PBW extensions, J. Algebr. Syst. 10 (1) (2022) 1–29.
  * [49] O. Hinchcliffe, Diffusion algebras, PhD Thesis, University of Sheffield, Sheffield (2005).
  * [50] C. Y. Hong, N. K. Kim, T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (3) (2000), 215–226.
  * [51] A. P. Isaev, P. N. Pyatov, V. Rittenberg, Diffusion algebras, J. Phys. A. 34 (29) (2001) 5815–5834.
  * [52] D. A. Jordan, Iterated Skew Polynomial Rings and Quantum Groups, J. Algebra 156 (1) (1993) 194–218.
  * [53] D. A. Jordan, Height one prime ideals of certain iterated skew polynomial rings, Math. Proc. Cambridge Philos. Soc. 114 (3) (1993) 407–425.
  * [54] D. A. Jordan, Finite-dimensional simple modules over certain iterated skew polynomial rings, J. Pure Appl. Algebra 98 (1) (1995) 45–55.
  * [55] D. A. Jordan, A simple localization of the quantized Weyl algebra, J. Algebra 174 (1) (1995) 267–281.
  * [56] D. A. Jordan, Down-Up Algebras and Ambiskew Polynomial Rings, J. Algebra 228 (1) (2000) 311–346.
  * [57] D. A. Jordan, I. Wells, Invariants for automorphisms of certain iterated skew polynomial rings, Proc. Edinb. Math. Soc. (2) 39(3) (1996) 461–472.
  * [58] V. G. Kac, Lie Superalgebras, Adv. Math. 26 (1) (1977) 8–96.
  * [59] V. G. Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 (1) (1977) 31–64.
  * [60] K. Kanakoglou, C. Daskaloyannis, Bosonisation and Parastatistics, in: S. Silvestrov, E. Paal, V. Abramov, A. Stolin (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer, Berlin, Heidelberg, 2009, pp. 207–218.
  * [61] A. Kandri-Rody, V. Weispfenning, Non-commutative Gröbner Bases in Algebras of Solvable Type, J. Symbolic Comput. 9 (1) (1990) 1–26.
  * [62] I. Kaplansky, Rings of Operators, Mathematics Lecture Notes Series, New York, Benjamin, 1965.
  * [63] E. Kirkman, I. M. Musson, D. S. Passman, Noetherian down-up algebras, Proc. Amer. Math. Soc. 127 (11) (1999) 3161–3167.
  * [64] J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (4) (1996) 289–300.
  * [65] R. S. Kulkarni, Irreducible Representations of Witten’s Deformations of $U(\mathfrak{sl}_{2})$, J. Algebra 214 (1) (1999) 64–91.
  * [66] L. Le Bruyn, Two remarks on Witten’s quantum enveloping algebra, Comm. Algebra 22 (3) (1994) 865–876.
  * [67] L. Le Bruyn, Conformal $\mathfrak{sl}_{2}$ enveloping algebras, Comm. Algebra 23 (4) (1995) 1325–1362.
  * [68] A. Leroy, J. Matczuk, On Induced Modules Over Ore Extensions, Comm. Algebra 32 (7) (2004) 2743–2766.
  * [69] O. Lezama, Computation of point modules of finitely semi-graded rings, Comm. Algebra 48 (2) (2020) 866-878.
  * [70] O. Lezama, J. P. Acosta, A. Reyes, Prime ideals of skew PBW extensions, Rev. Un. Mat. Argentina 56 (2) (2015) 39–55.
  * [71] O. Lezama, A. Reyes, Some Homological Properties of Skew PBW Extensions, Comm. Algebra 42 (3) (2014) 1200–1230.
  * [72] O. Lezama, H. Venegas, Center of skew PBW extensions, Internat. J. Algebra Comput. 30 (08) (2020) 1625–1650.
  * [73] M. Louzari, A. Reyes, Minimal prime ideals of skew PBW extensions over 2-primal compatible rings, Rev. Colombiana Mat. 54 (1) (2020) 39–63.
  * [74] M. Louzari, A. Reyes, Generalized Rigid Modules and Their Polynomial Extensions, in: M. Siles Molina, L. El Kaoutit, M. Louzari, L. Ben Yakoub, M. Benslimane (Eds.), Associative and Non-Associative Algebras and Applications, MAMAA 2018, Springer, Proceeding in Mathematics & Statistics, Vol. 311, Springer, Cham, 2020, pp. 147–158.
  * [75] G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (5) (2001) 2113–2123.
  * [76] G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2) (2003) 494–520.
  * [77] J. McConnell, J. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, New-York, American Mathematical Society (2001).
  * [78] I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43.
  * [79] A. Niño, M. C. Ramírez, A. Reyes, Associated prime ideals over skew PBW extensions, Comm. Algebra 48 (12) (2020) 5038–5055.
  * [80] A. Niño, A. Reyes, Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions, Algebra Discrete Math. 30 (2) (2020) 207–229.
  * [81] L. Ouyang, Extensions of nilpotent p.p. rings, Bull. Iranian Math. Soc. 36 (2) (2010) 169–184.
  * [82] L. Ouyang, G. F. Birkenmeier, Weak annihilator over extension rings, Bull. Malays. Math. Sci. Soc. 35 (2) (2012) 345–347.
  * [83] L. Ouyang, J. Liu, On weak $(\alpha,\delta)$-compatible rings, Int. J. Algebra 5 (26) (2011) 1283–1296.
  * [84] L. Ouyang, J. Liu, Weak associated primes over differential polynomial rings, Rocky Mountain J. Math. 42 (5) (2012) 1583–1600.
  * [85] L. Ouyang, J. Liu, Y. Xiang, Ore extensions of skew $\pi$-Armendariz rings, Bull. Iranian Math. Soc. 39 (2) (2013) 355–368.
  * [86] O. Ore, Theory of Non-Commutative Polynomials, Ann. of Math. (2) 34 (3) (1933) 480–508.
  * [87] P. N. Pyatov, R. Twarock, Construction of diffusion algebras, J. Math. Phys. 43 (6) (2002) 3268–3279.
  * [88] I. T. Redman, The Non-commutative Algebraic Geometry of some Skew Polynomial algebras, PhD Thesis, University of Wisconsin, Milwaukee (1996).
  * [89] I. T. Redman, The homogenization of the three dimensional skew polynomial algebras of type I, Comm. Algebra 27 (11) (1999) 5587–5602.
  * [90] A. Reyes, $\sigma$-PBW Extensions of Skew $\Pi$-Armendariz Rings, Far East J. Math. Sci. (FJMS) 103 (2) (2018) 401–428.
  * [91] A. Reyes, Armendariz modules over skew PBW extensions, Comm. Algebra 47 (3) (2019) 1248–1270.
  * [92] A. Reyes, C. Rodríguez, The McCoy Condition on Skew Poincaré-Birkhoff-Witt Extensions, Commun. Math. Stat. 9 (1) (2021) 1–21.
  * [93] A. Reyes, C. Sarmiento, On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras, Internat. J. Algebra Comput. (2022), https://doi.org/10.1142/S0218196722500242.
  * [94] A. Reyes, H. Suárez, A notion of compatibility for Armendariz and Baer properties over skew PBW extensions, Rev. Un. Mat. Argentina 59 (1) (2018) 157–178.
  * [95] A. Reyes, H. Suárez, Skew Poincaré-Birkhoff-Witt extensions over weak compatible rings, J. Algebra Appl. 19 (12) (2020) 2050225.
  * [96] A. Reyes, H. Suárez, Radicals and Köthe’s Conjecture for Skew PBW Extensions, Commun. Math. Stat. 9 (2) (2021) 119–138.
  * [97] A. Reyes, H. Suárez, Skew PBW extensions over symmetric rings, Algebra Discrete Math. 32 (1) (2021) 76–102.
  * [98] A. Reyes, Y. Suárez, On the ACCP in skew Poincaré-Birkhoff-Witt extensions, Beitr. Algebra Geom. 59 (4) (2018) 625–643.
  * [99] A. Rosenberg, Non-Commutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and Its Applications, Vol. 330, Dordrecht, Springer, 1995.
  * [100] R. C. Shock, Polynomial rings over finite dimensional rings, Pacific J. Math. 42 (1) (1972) 251–257.
  * [101] A. Seidenberg, Differential Ideals in Rings of Finitely Generated Type, Amer. J. Math. 89 (1) (1967) 22–42.
  * [102] W. M. Seiler, Involution. The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms Computat. Math, Vol. 24, Springer, 2010.
  * [103] S. P. Smith, A Class of Algebras Similar to the Enveloping of $\mathfrak{sl}(2)$, Trans. Amer. Math. Soc. 322 (1) (1991) 285–314.
  * [104] H. Suárez, A. Chacón, A. Reyes, On NI and NJ skew PBW extensions, Comm. Algebra (2022), https://doi.org/10.1080/00927872.2022.2028799.
  * [105] S. H. Sun, Noncommutative rings in which every prime ideal is contained in a unique maximal ideal, J. Pure Appl. Algebra 76 (2) (1991) 179–192.
  * [106] A. B. Tumwesigye, J. Richter, S. Silvestrov, Centralizers in PBW extensions, in: S. Silvestrov, A. Malyarenko, M. Rancic (Eds.), Algebraic Structures and Applications, SPAS 2017, Springer, Proceedings in Mathematics & Statistics, Vol. 317, Springer, 2020, pp. 469–490.
  * [107] E. Witten, Gauge Theories, Vertex Models, and Quantum Groups, Nuclear Phys. B 330 (2-3) (1990) 285–346.
  * [108] E. Witten, Quantization of Chern-Simons Gauge Theory with Complex Gauge Group, Comm. Math. Phys. 137 (1) (1991) 29-66.
  * [109] S. L. Woronowicz, Twisted $SU(2)$-Group, An Example of a Non-commutative Differential Calculus, Publ. Res. Inst. Math. Sci. 23 (1987) 117–181.