# On NI and NJ skew PBW extensions

Héctor Suárez  Universidad Pedagógica y Tecnológica de Colombia, Sede Tunja
Campus Universitario<EMAIL_ADDRESS>,  Andrés Chacón  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 professor Oswaldo Lezama for
his brilliant academic career at the
Universidad Nacional de Colombia - Sede Bogotá

###### Abstract.

We establish necessary or sufficient conditions to guarantee that skew
Poincaré-Birkhoff-Witt extensions are NI or NJ rings. Our results extend those
corresponding for skew polynomial rings and establish similar properties for
other families of noncommutative rings such as universal enveloping algebras
and examples of differential operators.

###### Key words and phrases:

Jacobson radical, Levitzki radical, NI ring, NJ ring, skew polynomial ring,
skew PBW extension

###### 2020 Mathematics Subject Classification:

16N20; 16N40; 16S36; 16S37; 16S38

The authors were supported by the research fund of Faculty of Science, Code
HERMES 52464, Universidad Nacional de Colombia - Sede Bogotá, Colombia.

##  1\. Introduction

Several kinds of rings are defined in terms of their set of nilpotent
elements. For example, a ring $R$ is called NI if its set $N(R)$ of nilpotent
elements coincides with its upper radical $N^{*}(R)$ (the sum of all its nil
ideals of $R$). If $N(R)=J(R)$, where $J(R)$ is the Jacobson radical of $R$,
then $R$ is called NJ. $R$ is said to be 2-primal if $N(R)$ is equal to the
prime radical $N_{*}(R)$ of $R$ (the intersection of all prime ideals of $R$).
$R$ is called weakly 2-primal if $N(R)$ coincides with its Levitzki radical
$L(R)$ (the sum of all locally nilpotent ideals of $R$).

The NI and NJ rings have recently been investigated by several authors. For
instance, Hwang et al. [16] studied the structure of NI rings related to
strongly prime ideals and showed that minimal strongly prime ideals can be
lifted in NI rings ([16], Theorem 2.3). They proved that for an NI ring $R$,
$R$ is weakly pm (every strongly prime ideal of $R$ is contained in a unique
maximal ideal of $R$) if and only if the topological space of maximal ideals
of $R$ is a retract of the topological space of strongly prime ideals of $R$,
or equivalently, if the topological space of strongly prime ideals of $R$ is
normal ([16], Theorem 3.7). Also, they proved that $R$ is weakly pm if and
only if $R$ is pm (every prime ideal of $R$ is contained in a unique maximal
ideal of $R$) when $R$ is a symmetric ring (that is, $rst=0$ implies $rts=0$,
where $r,s,t\in R$) ([16], Theorem 3.8).

Concerning skew polynomial rings (also known as Ore extensions) introduced by
Ore [35], Bergen and Grzeszczuk [5] studied the Jacobson radical of skew
polynomial rings of derivation type $R[x;\delta]$ when the base ring $R$ has
no nonzero nil ideals. They proved that if $R$ is an algebra with no nonzero
nil ideals satisfying the acc condition on right annihilators of powers, then
$J(R[x;\delta])=0$ ([5], Theorem 2). In the case that $R$ is a semiprime
algebra where every nonzero ideal contains a normalizing element, then
$J(R[x;\delta])=0$ ([5], Theorem 3). Related to this topic, Nasr-Isfahani [31]
gave necessary and sufficient conditions for a skew polynomial ring of
derivation type $R[x;\delta]$ to be semiprimitive when $R$ has no nonzero nil
ideals ([31], Corollary 2.2). He also proved that
$J(R[x;\delta])=N(R[x;\delta])=N(R)[x;\delta]$ if and only if $N(R)$ is a
$\delta$-ideal of $R$ (i.e., $N(R)$ is an ideal of $R$ and
$\delta(N(R))\subseteq N(R)$) and $N(R[x;\delta])=N(R)[x;\delta]$ ([31],
Proposition 2.7). Now, according to [31], Proposition 2.8, if $R[x;\delta]$ is
NI then
$J(R[x;\delta])=N(R[x;\delta])=N(R)[x;\delta]=N^{*}(R)[x;\delta]=N^{*}(R[x;\delta])$.

Later, Nasr-Isfahani [32] computed the Jacobson radical of an NI
$\mathbb{Z}$-graded ring $R=\bigoplus_{i\in\mathbb{Z}}R_{i}$. He showed that
$J(R)=N(R)$ if and only if $R$ is NI ring and $J(R)\cap R_{0}$ is nil ([32],
Theorem 2.4). He also proved that $R[x;\sigma]$ is NJ if and only if
$R[x;\sigma]$ is NI and $J(R[x;\sigma])\cap R\subseteq N(R)$ ([32], Corollary
2.5 (1)). For a skew polynomial ring of mixed type $R[x;\sigma,\delta]$, he
showed that $R[x;\sigma,\delta]$ is NI and $N(R)$ is $\sigma$-rigid (i.e,
$r\sigma(r)\in N(R)$ implies $r\in N(R)$, where $r\in R$) if and only if
$N(R)$ is a $\sigma$-invariant ideal of $R$ ($N(R)$ is an ideal and
$\sigma^{-1}(N(R))=N(R)$) and $N(R[x;\sigma,\delta])=N(R)[x;\sigma,\delta]$,
and equivalently, $N(R)$ is a $\sigma$-rigid ideal of $R$ and
$N^{*}(R[x;\sigma,\delta])=N^{*}(R)[x;\sigma,\delta]$ ([32], Theorem 3.1).

Jiang et al., [17] studied the relationship between NJ rings and some families
of rings. They investigated extensions as Dorroh, Nagata, and Jordan. For a
ring $R$ and an automorphism $\sigma$ of $R$, they proved that if $R$ is
weakly 2-primal $\sigma$-compatible (following Annin [1], Definition 2.1, $R$
is said to be $\sigma$-compatible if for each $a,b\in R$, $ab=0$ if and only
if $a\sigma(b)=0$), then $R[x;\sigma]$ is NJ ([17], Theorem 3.10 (1)), and if
$R$ is a weakly 2-primal $\delta$-compatible ring ($R$ is said to be
$\delta$-compatible if for each $a,b\in R$, $ab=0$ implies $a\delta(b)=0$
([1], Definition 2.1)), then $R[x;\delta]$ is NJ ([17], Theorem 3.12 (1)).
Moreover, they considered some topological conditions for NJ rings and showed
relations between algebraic and topological notions ([17], Section 4).

Han et al., [11] showed that if $R$ is an NI ring, $a,b\in R$, and $ab\in
Z(R)$, the center of $R$, then $ab-ba\in N^{*}(R)$, and there exists $l\geq 1$
such that $(ab)^{n}=(ba)^{n}$, for every $n\geq l$ ([11], Theorem 1.3 (3)).
They also proved that for the ideal $I$ of $R$ generated by the subset $\\{ab-
ba\mid a,b\in R,\ {\rm such\ that}\ ab\in Z(R)\\}$, if $R$ is NI then $I$ is
nil and $R/I$ is an Abelian (i.e., the idempotents of $R$ are central elements
of $R$) NI ring ([11], Theorem 1.3 (5)).

With respect to the objects of interest in this paper, the skew Poincaré-
Birkhoff-Witt (PBW) extensions, these were defined by Gallego and Lezama [8]
with the aim of generalizing families of noncommutative rings as PBW
extensions introduced by Bell and Goodearl [3], skew polynomial rings (of
injective type) defined by Ore [35], and others as solvable polynomial rings,
diffusion algebras, some types of Auslander-Gorenstein rings, some Calabi-Yau
and skew Calabi-Yau algebras, some Artin-Schelter regular algebras, some
Koszul algebras, and others (see [7] or [46] for a detailed reference to each
of these families). Several ring and theoretical properties of skew PBW
extensions have been studied by different authors (see Artamonov [2], Fajardo
et al., [7], Hamidizadeh et al., [10], Hashemi et al., [12], [14], Lezama et
al., [18], [21], [23], [24], [27], Tumwesigye et al., [47], and Zambrano
[48]). In particular, the 2-primal property for these objects has been
investigated by Hashemi et al., [13], Louzari and Reyes [28], and the authors
[42]. Nevertheless, NI and NJ properties have not been studied for skew PBW
extensions, so it is a natural task to find necessary or sufficient conditions
under which they can be NI and NJ. This is the objective of the paper, and
therefore our results contribute to the study of ideals and radicals of skew
PBW extensions that has been partially carried out (see [13], [22], [28],
[33], [34], and [39]), and establish ring-theoretical properties for
noncommutative rings not considered in the literature. As a matter of fact, we
generalize some results appearing in Jiang et al., [17], and Nasr-Isfahani
[31] and [32].

The paper is organized as follows. In Section 2, we recall definitions and
properties needed for the rest of the paper. Section 3 contains the results
about the NI property for skew PBW extensions. More exactly, for
$A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ a skew PBW extension over a ring
$R$, we prove the following results: if $R$ is weak
$(\Sigma,\Delta)$-compatible, then $R$ is NI if and only if $A$ is NI (Theorem
3.3). If $A$ is of derivation type over $R$, then $A$ is NI if and only if
$N(R)$ is a $\Delta$-invariant ideal of $R$ and $N(A)=N(R)\langle
x_{1},\dots,x_{n}\rangle$ (Proposition 3.7). $A$ is NI and $N(R)$ is
$\Sigma$-rigid if and only if $N(R)$ is a $\Sigma$-ideal of $R$ and
$N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$, and equivalently, $N(R)$ is a
$\Sigma$-rigid ideal of $R$ and $N^{*}(A)=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle$ (Theorem 3.8). Next, in Section 4, results about NJ
property and their relations with NI property for skew PBW extensions are
presented. We prove that if $A$ is a graded skew PBW extension over an
$\mathbb{N}$-graded ring $R=\bigoplus_{n\in\mathbb{N}}R_{n}$, then $A$ is NJ
if and only if $A$ is NI and $J(A)\cap R_{0}$ is a nil ideal (Theorem 4.3). If
furthermore $A$ is connected, then $A$ is NJ if and only if $A$ is NI
(Corollary 4.4). If $A$ is quasi-commutative, bijective and $R$ is a weakly
2-primal weak $\Sigma$-compatible ring, then $A$ is NJ (Theorem 4.7). If $A$
is of derivation type, then $A$ is NI if and only if $A$ is NJ, and
equivalently, $R$ is NI and $N(A)=N(R)\langle x_{1},\dotsc,x_{n}\rangle$ if
and only if $R$ is NI and $N^{*}(A)=N^{*}(R)\langle x_{1},\dotsc,x_{n}\rangle$
(Proposition 4.13). Finally, if $A$ is quasi-commutative, then $A$ is NJ and
$N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$ if and only if $N(R)$ is a
$\Sigma$-ideal of $R$ and $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$, or
equivalently, $A$ is NI and $N(R)$ is $\Sigma$-rigid if and only if $N(R)$ is
$\Sigma$-rigid ideal of $R$ and $N^{*}(A)=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle$ (Proposition 4.14).

Finally, we present some ideas for a future work.

##  2\. Preliminaries

Throughout the paper, the term ring means an associative ring with identity
not necessarily commutative. For a ring $R$, as we saw above, we fix the
following notation: $N_{*}(R)$ is its prime radical, $N^{*}(R)$ is its upper
radical, $N(R)$ is its set of nilpotent elements, $J(R)$ is its Jacobson
radical, and $L(R)$ is its Levitzki radical. It is well-known that the
following relations hold: $N_{*}(R)\subseteq L(R)\subseteq N^{*}(R)\subseteq
N(R)$ and $N_{*}(R)\subseteq L(R)\subseteq N^{*}(R)\subseteq J(R)$ (see [19],
[30] or [31], for more details).

$R$ is called _nil-semisimple_ if it has no nonzero nil ideals. Recall that
nil-semisimple rings are semiprime ([16], p. 187). If $P$ is a prime ideal of
$R$, $P$ is called _completely prime_ if $R/P$ is a domain, and $P$ is said to
be _strongly prime_ if $R/P$ is nil-semisimple. Note that maximal ideals and
completely prime ideals are strongly prime; any strongly prime ideal contains
a minimal strongly prime ideal; $N^{*}(R)$ is the unique maximal nil ideal of
$R$; $N^{*}(R)=\\{a\in R\mid RaR\text{ is a nil ideal of }R\\}=\bigcap\\{P\mid
P\text{ is a strongly prime ideal of }R\\}$ (see [16] or [19] for more
details).

For a ring $R$, 2-primal implies weakly 2-primal, whence NI and NJ are
examples of NI rings. $R$ is reduced (i.e., without nonzero nilpotent
elements) if and only if $R$ is nil-semisimple and NI, or equivalently, $R$ is
semiprime and 2-primal ([16], p. 187). As we can check, $R$ is 2-primal if and
only if $N_{*}(R)=N^{*}(R)=N(R)$. Shin [44], Proposition 1.11, proved that the
set of nilpotent elements of a ring $R$ coincides with the prime radical if
and only if every minimal prime ideal of $R$ is completely prime. Thus, $R$ is
2-primal if and only if every minimal prime ideal of $R$ is completely prime,
or equivalently, $R/N_{*}(R)$ is reduced.

A ring $R$ is said to be _semicommutative_ if for every pair of elements
$a,b\in R$, we have that $ab=0$ implies $aRb=0$. Shin [44], Lemma 1.2 and
Theorem 1.5, established that semicommutative rings are 2-primal, and hence
semi-commutative rings are NI. Note that domains are reduced rings, reduced
rings are symmetric, symmetric rings are reversible ($R$ is said to be
reversible if $ab=0$ implies $ba=0$, where $a,b\in R$), and reversible rings
are semicommutative, but the converses are not true in general (see [30]). A
ring is called _right_ (left) duo if every right (left) ideal of it is two-
sided. Shin [44], Lemma 1.2, showed that right (left) duo rings are
semicommutative. Therefore, NI rings contain several families of rings such as
domains, reduced rings, symmetric rings, semi-commutative rings, reversible
rings, one-sided duo rings, 2-primal rings and NJ rings (for more details, see
[29]). Köthe’s conjecture (the upper nilradical contains every nil left ideal)
holds for NI rings (Hwang et al., [16], p. 192).

Equivalent definitions for NI rings are presented in Proposition 2.1.

######  Proposition 2.1 ([16], Lemma 2.1).

For a ring $R$, the following conditions are equivalent.

  1. (i)

$R$ is NI.

  2. (ii)

$N(R)$ is an ideal.

  3. (iii)

Every subring (possibly without identity) of $R$ is NI.

  4. (iv)

Every minimal strongly prime ideal of $R$ is completely prime.

  5. (v)

$R/N^{*}(R)$ is a reduced ring.

  6. (vi)

$R/N^{*}(R)$ is a symmetric ring.

Some examples of NJ rings are nil rings, division rings, Boolean rings,
commutative Jacobson rings, commutative affine algebras over a field
$\mathbb{K}$, semi-Abelian $\pi$-regular rings, locally finite Abelian rings
(Jiang et al., [17], Example 2.5). Every reduced ring is NJ ([17], Proposition
2.11). Note that $R=\mathbb{Z}[[x]]$ is a domain and hence NI, with
$N(R)=\\{0\\}$, but $J(\mathbb{Z}[[x]])=x\mathbb{Z}[[x]]\neq\\{0\\}$ and so
$R$ is not an NJ ring ([17], Example 2.2). This example shows that NI are not
included in NJ rings.

Next, we present the objects of interest in this paper with some of their
ring-theoretical notions. The symbol $\mathbb{N}$ denotes the set of natural
numbers including the zero element.

######  Definition 2.2 ([8], Definition 1).

Let $R$ and $A$ be rings. $A$ is called a skew PBW extension over $R$, 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 identity element.

  2. (ii)

there exist finitely many elements $x_{1},\dots,x_{n}\in A$ such that $A$ is a
left free $R$-module, with basis the set of standard monomials

${\rm Mon}(A):=\\{x^{\alpha}:=x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}\mid\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}\\}$.

Moreover, $x^{0}_{1}\cdots x^{0}_{n}:=1\in{\rm Mon}(A)$.

  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 $1\leq i,j\leq n$, there exists $d_{i,j}\in R\ \backslash\ \\{0\\}$ such
that

(2.1) $x_{j}x_{i}-d_{i,j}x_{i}x_{j}\in R+Rx_{1}+\cdots+Rx_{n}.$

Note that every element $f\in A\ \setminus\ \\{0\\}$ can be expressed uniquely
as $f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}$, with $a_{i}\in R\setminus\\{0\\}$
and $X_{i}\in{\rm Mon}(A)$, $0\leq i\leq m$ ([8], Remark 2). For
$X=x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\in{\rm Mon}(A)$,
$\deg(X)=|\alpha|:={\alpha_{1}}+\cdots+{\alpha_{n}}$.

######  Proposition 2.3 ([8], Proposition 3).

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension over
$R$. For each $1\leq i\leq n$, 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 every $r\in R$.

From now on $\Sigma:=\\{\sigma_{1},\dotsc,\sigma_{n}\\}$ and
$\Delta:=\\{\delta_{1},\dotsc,\delta_{n}\\}$. We say that $(\Sigma,\Delta)$ is
a _system of endomorphisms and $\Sigma$-derivations_ of $R$ with respect to
$A$. For $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$,
$\sigma^{\alpha}:=\sigma_{1}^{\alpha_{1}}\circ\cdots\circ\sigma_{n}^{\alpha_{n}}$,
$\delta^{\alpha}:=\delta_{1}^{\alpha_{1}}\circ\cdots\circ\delta_{n}^{\alpha_{n}}$,
where $\circ$ denotes composition of functions.

Following [8], Definition 4, skew PBW extension $A$ is called bijective if
$\sigma_{i}$ is bijective and $d_{i,j}$ is invertible for any $1\leq i<j\leq
n$. $A$ is called quasi-commutative if the conditions (iii) and (iv) in
Definition 2.2 are replaced by the following:

  1. (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 $x_{i}r=c_{i,r}x_{i}$;

  2. (iv’)

for any $1\leq i,j\leq n$, there exists $d_{i,j}\in R\ \backslash\ \\{0\\}$
such that $x_{j}x_{i}=d_{i,j}x_{i}x_{j}$.

If $\sigma_{i}$ is the identity map of $R$, for each $i=1,\dotsc,n$, (we write
$\sigma_{i}={\rm id}_{R}$), we say that $A$ is a skew PBW extension of
derivation type. Similarly, if $\delta_{i}=0$, for each $\delta_{i}\in\Delta$,
then $A$ is said to be a skew PBW extension of endomorphism type.

######  Remark 2.4.

  1. (i)

From Definition 2.2 (iv), it is clear that skew PBW extensions are more
general than iterated skew polynomial rings (cf. [9]). For example, universal
enveloping algebras of finite dimensional Lie algebras and some 3-dimensional
skew polynomial algebras in the sense of Bell and Smith [4] (see also
Rosenberg [43]) cannot be expressed as iterated skew polynomial rings but are
skew PBW extensions. For quasi-commutative skew PBW extensions, these are
isomorphic to iterated Ore extensions of endomorphism type ([26], Theorem
2.3).

  2. (ii)

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 2.2 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.

  3. (iii)

PBW extensions introduced by Bell and Goodearl [3] are particular examples of
skew PBW extensions (see [8], Section 1 for a detailed description). By
definition, the first objects satisfy the relation
$x_{i}r=rx_{i}+\delta_{i}(r)$, so that these structures are examples of skew
PBW extensions of derivation type. As examples, we mention the following: the
enveloping algebra of a finite-dimensional Lie algebra; any differential
operator ring $R[\theta_{1},\dotsc,\theta_{1};\delta_{1},\dotsc,\delta_{n}]$
formed from commuting derivations $\delta_{1},\dotsc,\delta_{n}$; differential
operators introduced by Rinehart; twisted or smash product differential
operator rings, and others (for more details, see [3], p. 27).

From Definition 2.2, it follows that skew PBW extensions are not
$\mathbb{N}$-graded ring in a non-trivial sense. With this in mind,
Proposition 2.5 allows to define a subfamily of these extensions, the graded
skew PBW extensions (Definition 2.6) that were introduced by the first author
in [45]. Before presenting the definition, we recall that if
$R=\bigoplus_{p\in\mathbb{N}}R_{p}$ and $S=\bigoplus_{p\in\mathbb{N}}S_{p}$
are $\mathbb{N}$-graded rings, then a map $\varphi:R\to S$ is called _graded_
if $\varphi(R_{p})\subseteq S_{p}$, for each $p\in\mathbb{N}$. For
$m\in\mathbb{N}$, $R(m):=\bigoplus_{p\in\mathbb{N}}R(m)_{p}$, where
$R(m)_{p}:=R_{p+m}$.

######  Proposition 2.5 ([45], Proposition 2.7(ii)).

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a bijective skew PBW
extension over an $\mathbb{N}$-graded algebra $R=\bigoplus_{m\geq 0}R_{m}$. If
the following conditions hold:

  1. (i)

$\sigma_{i}$ is a graded ring homomorphism and $\delta_{i}:R(-1)\to R$ is a
graded $\sigma_{i}$-derivation, for all $1\leq i\leq n$, and

  2. (ii)

$x_{j}x_{i}-d_{i,j}x_{i}x_{j}\in R_{2}+R_{1}x_{1}+\cdots+R_{1}x_{n}$, as in
(2.1) and $d_{i,j}\in R_{0}$,

then $A$ is an $\mathbb{N}$-graded algebra with graduation given by
$A=\bigoplus_{p\geq 0}A_{p}$, where for $p\geq 0$, $A_{p}$ is the
$\mathbb{K}$-space generated by the set

$\Bigl{\\{}r_{t}x^{\alpha}\mid t+|\alpha|=p,\ r_{t}\in R_{t}\text{ and
}x^{\alpha}\in{\rm Mon}(A)\Bigr{\\}}.$

######  Definition 2.6 ([45], Definition 2.6).

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a bijective skew PBW
extension over an $\mathbb{N}$-graded algebra $R=\bigoplus_{m\geq 0}R_{m}$. If
$A$ satisfies the conditions (i) and (ii) established in Proposition 2.5, then
we say that $A$ is a graded skew PBW extension over $R$.

We present some remarkable examples of graded skew PBW extensions.

######  Example 2.7.

The Jordan plane, homogenized enveloping algebras, and some classes of
diffusion algebras ([45], Examples 2.9). Graded Clifford algebras defined by
Le Bruyn [20] are also examples of graded skew PBW extensions if we assume the
condition of PBW basis. Let us see the details.

Following Cassidy and Vancliff [6], let $\mathbb{K}$ be an algebraically
closed field such that ${\rm char}(\mathbb{K})\neq 2$ and let
$M_{1},\dots,M_{n}\in\mathbb{M}_{n}(\mathbb{K})$ be symmetric matrices of
order $n\times n$ with entries in $\mathbb{K}$. A _graded Clifford algebra_
$\mathcal{A}$ is a $\mathbb{K}$-algebra on degree-one generators
$x_{1},\dots,x_{n}$ and on degree-two generators $y_{1},\dots,y_{n}$ with
defining relations given by:

  1. (i)

$x_{i}x_{j}+x_{j}x_{i}=\sum_{k=1}^{n}(M_{k})_{ij}y_{k}$ for all
$i,j=1,\dots,n$;

  2. (ii)

$y_{k}$ central for all $k=1,\dots,n$.

Note that the commutative polynomial ring $R=\mathbb{K}[y_{1},\dots,y_{n}]$ is
an $\mathbb{N}$-graded algebra where $R_{0}=\mathbb{K},R_{1}=\\{0\\}$,
$y_{1},\dots,y_{n}\in R_{2}$, and $R_{i}=\\{0\\}$, for $i\geq 3$. If we
suppose that the set $\\{x_{1}^{a_{1}}\dotsb x_{n}^{a_{n}}\mid
a_{i}\in\mathbb{N},i=1,\dotsc,n\\}$ is a left PBW $R$-basis for $\mathcal{A}$,
then the graded Clifford algebra $\mathcal{A}$ is a graded skew PBW extension
over the connected algebra $R$, that is, $\mathcal{A}\cong\sigma(R)\langle
x_{1},\dotsc,x_{n}\rangle$. Indeed, from the relations (i) and (ii) above, it
is clear that $\sigma_{i}={\rm id}_{R}$, $\delta_{i}=0$, $d_{i,j}=-1\in
R_{0}$, for $1\leq i,j\leq n$, and $\sum_{k=1}^{n}(M_{k})_{ij}y_{k}\in R_{2}$,
where $d_{i,j}$ is given as in expression (2.1). In this way, $\mathcal{A}$ is
a bijective skew PBW extension that satisfies the conditions of Proposition
2.5.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension over
a ring $R$, and consider the sets $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$
and $\Delta:=\\{\delta_{1},\dots,\delta_{n}\\}$ as above. An ideal $I$ of $R$
is called $\Sigma$-_ideal_ if $\sigma_{i}(I)=I$, for each $1\leq i\leq n$.
From Lezama et al., [22], Definition 2.1, $I$ is called $\Sigma$-_invariant_
if $\sigma_{i}(I)\subseteq I$, and it is called $\Delta$-_invariant_ if
$\delta_{i}(I)\subseteq I$, for $1\leq i\leq n$. If $I$ is both $\Sigma$ and
$\Delta$-invariant, we say that $I$ is $(\Sigma,\Delta)$-_invariant_. From
Reyes [36], Definition 3.2, $R$ is $\Sigma$-_rigid_ if $r\sigma^{\alpha}(r)=0$
implies $r=0$, where $r\in R$ and $\alpha\in\mathbb{N}^{n}$. A subset
$S\subseteq R$ is $\Sigma$-_rigid_ if $r\sigma^{\alpha}(r)\in S$ implies $r\in
S$, where $r\in R$ and $\alpha\in\mathbb{N}^{n}$. Following Hashemi et al.,
[13], for $S\subseteq R$, we denote the set of all elements of $A$ with
coefficients in $S$ by $S\langle x_{1},\dots,x_{n}\rangle$. Definitions of
$(\Sigma,\Delta)$-ideal $I$ and $I\langle x_{1},\dots,x_{n}\rangle=I\langle
x_{1},\dots,x_{n};\Sigma,\Delta\rangle$ introduced by Hashemi et al., [13] are
the same as those of $(\Sigma,\Delta)$-invariant ideal $I$ and $IA$,
respectively, considered in [22]. Note that the terminology used by Nasr-
Isfahani [32] is different, since for $\sigma$ an endomorphism of $R$,
$\delta$ a $\sigma$-derivation of $R$, and $I$ an ideal of $R$, $I$ is called
$\sigma$-invariant if $\sigma^{-1}(I)=I$, and it is called a $\delta$-ideal if
$\delta(I)\subseteq I$; this same terminology is used in Nasr-Isfahani [31],
[32]. Finally, from [22], Proposition 2.6 (i), if $I$ is a
$(\Sigma,\Delta)$-invariant ideal of $R$, then $IA=I\langle
x_{1},\dots,x_{n};\Sigma,\Delta\rangle=I\langle x_{1},\dots,x_{n}\rangle$ is
an ideal of $A$, $IA\cap R=I$ and $IA$ is proper if and only if $I$ is proper.

Next, we present different ring-theoretical notions that have been defined for
skew PBW extensions and extend corresponding notions for skew polynomial rings
(for more details, see [38]).

######  Definition 2.8.

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

  1. (i)

([38], Definition 3.3) $R$ is called a $\Sigma$-_skew Armendariz_ ring if for
elements $f=\sum_{i=0}^{m}a_{i}X_{i}$ and $g=\sum_{j=0}^{t}b_{j}Y_{j}$ in $A$,
the equality $fg=0$ implies $a_{i}\sigma^{\alpha_{i}}(b_{j})=0$, for all
$0\leq i\leq m$ and $0\leq j\leq t$, where $\alpha_{i}={\rm exp}(X_{i})$. $R$
is called a _weak $\Sigma$-skew Armendariz_ ring if for elements
$f=\sum_{i=0}^{n}a_{i}x_{i}$ and $g=\sum_{j=0}^{n}b_{j}x_{j}$ in $A$
$(x_{0}:=1)$, the equality $fg=0$ implies $a_{i}\sigma_{i}(b_{j})=0$, for all
$0\leq i,j\leq n$ $(\sigma_{0}:={\rm id}_{R})$.

  2. (ii)

([12], Definition 3.2; [39], Definition 3.1) $R$ 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 $a,b\in R$, $ab=0$ 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.

  3. (iii)

([41], 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, $R$ is called weak $(\Sigma,\Delta)$\- compatible.

Several relations between above skew Armendariz notions and different examples
can be found in [37], Example 3.4, and [40], Section 5.

######  Remark 2.9.

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

  1. (i)

$I$ is $\Sigma$-invariant if and only if $\sigma^{\alpha}(I)\subseteq I$, for
every $\alpha\in\mathbb{N}^{n}$.

  2. (ii)

$I$ is $\Delta$-invariant if and only if $\delta^{\alpha}(I)\subseteq I$, for
every $\alpha\in\mathbb{N}^{n}$.

  3. (iii)

$I$ is $\Sigma$-ideal if and only if $\sigma_{i}^{-1}(I)=I$, for each $1\leq
i\leq n$. Also, if for each $1\leq i\leq n$, $\sigma_{i}^{-1}(I)=I$ then
$\sigma^{-\alpha}(I)=I$, for $\alpha\in\mathbb{N}^{n}$. Therefore, the
definition of $\Sigma$-ideal given in this paper coincides with the definition
of $\Sigma$-invariant ideal given by Hashemi et al., [13], Definition 3.1, and
the definition of $\alpha$-invariant ideal presented by Nasr-Isfahani [32], p.
5116.

##  3\. NI skew PBW extensions

This section contains the original results of the paper about NI property for
skew PBW extensions. We start with Proposition 3.1 which follows directly from
[28], Theorem 3.9, and [42], Proposition 4.4. Recall that a ring $R$ is
locally finite if every finite subset in $R$ generates a finite semigroup
multiplicatively.

######  Proposition 3.1.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension over
$R$. If $R$ satisfies one of the following conditions,

  1. (i)

$R$ is 2-primal and $(\Sigma,\Delta)$-compatible, or

  2. (ii)

$R$ is locally finite, $(\Sigma,\Delta)$-compatible and $\Sigma$-skew
Armendariz,

then $A$ is an NI ring.

######  Proposition 3.2.

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is an NI skew PBW extension
over $R$, then $R$ is an NI ring, $N(R)\subseteq J(R)$, $N(A)\subseteq J(A)$,
and therefore $N(R)\subseteq J(A)$.

###### Proof.

From Definition 2.2 (i) we know that $R$ is a subring of $A$. Since $A$ is NI,
Proposition 2.1 implies that $R$ is NI. Now, since the Jacobson radical of a
ring contains the nil ideals, by Proposition 2.1 we have that $N(A)$ and
$N(R)$ are nil ideals of $A$ and $R$, respectively. Hence, $N(R)\subseteq
J(R)$ and $N(A)\subseteq J(A)$. As $N(R)\subseteq N(A)$, then $N(R)\subseteq
J(A)$. ∎

From Propositions 3.1 and 3.2 we deduce that if $R$ is locally finite,
$(\Sigma,\Delta)$-compatible and $\Sigma$-skew Armendariz, then $R$ is NI.
This result has been proved by Reyes and Suárez [42], Theorem 4.3.

Since weak $(\Sigma,\Delta)$-compatible rings are more general than
$(\Sigma,\Delta)$-compatible rings, and NI rings are more general than
2-primal rings, the following theorem generalizes [13], Theorem 4.11, and some
other results of [13] and [28] formulated for skew PBW extensions over
2-primal $(\Sigma,\Delta)$-compatible rings.

######  Theorem 3.3.

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a skew PBW extension over
a weak $(\Sigma,\Delta)$-compatible ring $R$, then $R$ is NI if and only if
$A$ is NI.

###### Proof.

Suppose that $R$ is an NI ring. Let us first see that $N(R)$ is a
$(\Sigma,\Delta)$-invariant. By Proposition 2.1, $N(R)$ is an ideal of $R$.
For a fixed $i$, if $\sigma_{i}(r)\in\sigma_{i}(N(R))$, where $r\in N(R)$,
then $\sigma_{i}(r^{k})=\sigma_{i}(r)^{k}=0$, for some positive integer $k$.
Thus $\sigma_{i}(r)\in N(R)$, i.e., $\sigma_{i}(N(R))\subseteq N(R)$. Now, for
$r\in N(R)$, $\delta_{i}(r)\in N(R)$, since $R$ is weak
$(\Sigma,\Delta)$-compatible, whence $\delta_{i}(N(R))\subseteq N(R)$. Since
$N(R)$ is a $(\Sigma,\Delta)$-invariant ideal or $R$, [22], Proposition 2.6
(i), implies that $N(R)\langle x_{1},\dots,x_{n}\rangle$ is an ideal of $A$.

Let us show that $N(R)\langle x_{1},\dots,x_{n}\rangle=N(A)$. From [41],
Theorem 4.6, $f=\sum_{i=0}^{t}a_{i}X_{i}\in N(R)\langle
x_{1},\dots,x_{n}\rangle$ if and only if $a_{i}\in N(R)$, for $0\leq i\leq t$,
if and only if $f=\sum_{i=0}^{t}a_{i}X_{i}\in N(A)$. Therefore, $N(A)$ is an
ideal of $A$, and by Proposition 2.1, we have that $A$ is an NI ring.

Conversely, if $A$ is an NI ring, then by Proposition 3.2 we have that $R$ is
an NI ring. ∎

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a skew PBW extension over
a $(\Sigma,\Delta)$-compatible ring $R$, then $N(R)$ is $\Sigma$-rigid.
Indeed, for $r\in R$ and $\alpha\in\mathbb{N}^{n}$, if $r\sigma^{\alpha}(r)\in
N(R)$, then [40], Lemma 2, implies that $r^{2}\in N(R)$, whence $r\in N(R)$.

For the next result, recall that a ring $R$ is said to be Dedekind finite if
$ab=1$ implies $ba=1$, where $a,b\in R$.

######  Proposition 3.4.

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is an NI skew PBW extension
over $R$, then:

  1. (i)

$N(R)$ and $N(A)$ are completely semiprime.

  2. (ii)

$d_{i,j}$ in Definition 2.2 are units. In general, left (resp. right)
invertible elements in $A$ are units.

###### Proof.

  1. (i)

Since $A$ and $R$ are NI rings, then $N(R)$ is an ideal of $R$ and $N(A)$ is
an ideal of $A$. If $r^{2}\in N(R)$ then $(r^{2})^{k}=r^{2k}=0$, for some
positive integer $k$, and so $r\in N(R)$. Analogously, if $f^{2}\in N(A)$ then
$f\in N(A)$. Therefore, $R$ and $A$ are completely semiprime.

  2. (ii)

By [8], Remark 2 (iii), for every $1\leq i<j\leq n$, $d_{i,j}$ has a left
inverse and $d_{j,i}$ has a right inverse. In this way, there exist elements
$r$, $s\in R$ such that $rd_{i,j}=1=d_{j,i}s$. Since $R$ is an NI ring, then
by [16], Proposition 2.7 (1), $R$ is Dedekind finite, whence
$d_{i,j}r=1=sd_{j,i}$. Now, if $f\in A$ is left (resp. right) invertible, then
$gf=1$ (resp. $fh=1$), for some $g,h\in A$. Since $A$ is NI, $A$ is Dedekind
finite and so $fg=1$ (resp. $hf=1$).

∎

######  Remark 3.5.

In [37], Theorem 3.14, it was presented a relation between skew PBW extensions
of endomorphism type and the notion of Dedekind finite by considering a skew
notion of McCoy ring.

######  Proposition 3.6.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension of
derivation type and $I\subseteq R$. Then $I\langle x_{1},\dots,x_{n}\rangle$
is an ideal of $A$ if and only if $I$ is a $\Delta$-invariant (and therefore
$(\Sigma,\Delta)$-invariant) ideal of $R$.

###### Proof.

If $I\langle x_{1},\dots,x_{n}\rangle$ is an ideal of $A$, then $I$ is an
ideal of $R$. Let $\delta_{i}(r)\in\delta_{i}(I)$ such that $r\in I$. Then
$x_{i}r=rx_{i}+\delta_{i}(r)\in I\langle x_{1},\dots,x_{n}\rangle$, for each
$1\leq i\leq n$. As $-rx_{i}\in I\langle x_{1},\dots,x_{n}\rangle$ then
$\delta_{i}(r)\in I\langle x_{1},\dots,x_{n}\rangle$; in particular,
$\delta_{i}(r)\in I$. This means that $I$ is a $\Delta$-invariant ideal. Since
$A$ is of derivation type, $I$ is a $\Sigma$-invariant ideal.

The converse follows from [22], Proposition 2.6 (i). ∎

######  Proposition 3.7.

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a skew PBW extension of
derivation type over $R$, then the following assertions hold:

  1. (i)

$N(R)$ is $\Sigma$-rigid.

  2. (ii)

If $A$ is NI then $N(R)$ and $N^{*}(R)$ are $\Sigma$-rigid ideals.

  3. (iii)

For every completely prime $P$ of $A$, $P\cap R$ is a completely prime ideal
of $R$.

  4. (iv)

$A$ is NI if and only if $N(R)$ is a $\Delta$-invariant ideal of $R$ and
$N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$.

###### Proof.

  1. (i)

If $r\in R$ satisfies $r^{2}=r\sigma^{\alpha}(r)\in N(R)$, then $r\in N(R)$,
i.e., $N(R)$ is $\Sigma$-rigid.

  2. (ii)

If $A$ is NI, then $R$ is NI and therefore $N(R)$ is an ideal of $R$. By (i),
$N(R)$ is $\Sigma$-rigid, so $N(R)$ is a $\Sigma$-rigid ideal. Since
$N(R)=N^{*}(R)$, then $N^{*}(R)$ is a $\Sigma$-rigid ideal.

  3. (iii)

From [34], Theorem 1, for every completely prime ideal $P$ of $A$, $P\cap R$
is a completely prime ideal of $R$.

  4. (iv)

If $A$ is NI, then by Proposition 3.2 $R$ is NI, and so $N(R)$ is an ideal of
$R$. Let $\delta_{i}(r)\in\delta_{i}(N(R))$ such that $r\in N(R)\subseteq
N(A)$. Since $A$ is NI, $N(A)$ is an ideal of $A$ and
$x_{i}r=rx_{i}+\delta_{i}(r)\in N(A)$, for each $1\leq i\leq n$. As
$-rx_{i}\in N(A)$, then $\delta_{i}(r)\in N(A)\cap R=N(R)$, which means that
$N(R)$ is a $\Delta$-invariant ideal. From [13], Proposition 4.1,
$N(A)\subseteq N(R)\langle x_{1},\dots,x_{n}\rangle$. For the another
inclusion, let $f=a_{0}+a_{1}X_{1}+\dots+a_{t}X_{t}\in N(R)\langle
x_{1},\dots,x_{n}\rangle$, with $a_{i}\in N(R)\subseteq N(A)$, for $0\leq
i\leq t$. Since $N(A)$ is an ideal of $A$,
$a_{0},a_{1}X_{1},\dots,a_{t}X_{t}\in N(A)$, and so
$f=a_{0}+a_{1}X_{1}+\dots+a_{t}X_{t}\in N(A)$.

Conversely, if $N(R)$ is a $\Delta$-invariant ideal, Proposition 3.6
guarantees that $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$ is an ideal of
$A$. Proposition 2.1 implies that $A$ is an NI ring.

∎

The following result is one of the most important in the paper. This extends
Nasr-Isfahani [32], Theorem 3.1.

######  Theorem 3.8.

If $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ is a skew PBW extension over
a ring $R$, then the following statements are equivalent:

  1. (i)

$A$ is NI and $N(R)$ is $\Sigma$-rigid.

  2. (ii)

$N(R)$ is a $\Sigma$-ideal of $R$ and $N(A)=N(R)\langle
x_{1},\dots,x_{n}\rangle$.

  3. (iii)

$N(R)$ is a $\Sigma$-rigid ideal of $R$ and $N^{*}(A)=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle$.

###### Proof.

(i) $\Rightarrow$ (ii) Suppose that $A$ is NI and $N(R)$ is $\Sigma$-rigid. By
Proposition 3.2, $R$ is NI and so $N(R)$ is an ideal of $R$. From Proposition
2.3 we know that every $\sigma_{i}$ is injective, whence $r\in N(R)$ if and
only if $r^{k}=0$, for some positive integer $k$, if and only if
$\sigma_{i}(r^{k})=(\sigma_{i}(r))^{k}=0$ if and only if $\sigma_{i}(r)\in
N(R)$, and equivalently, $r\in\sigma_{i}^{-1}(N(R))$, which shows that $N(R)$
is a $\Sigma$-ideal of $R$.

With the aim of showing that $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$,
before consider the following facts.

Note that $N(R)$ is $\Delta$-invariant; indeed, if $r\in N(R)\subseteq N(A)$,
then $x_{i}r=\sigma_{i}(r)x_{i}+\delta_{i}(r)\in N(A)$. Since $N(R)$ is a
$\Sigma$-ideal, then $\sigma_{i}(r)\in N(R)\subseteq N(A)$, and so
$\sigma_{i}(r)x_{i}\in N(A)$, which implies that $\delta_{i}(r)\in N(A)$, that
is, $\delta_{i}(r)\in N(R)$. By [22], Proposition 2.2 (i), the system of
endomorphisms and $\Sigma$-derivations $(\Sigma,\Delta)$ induces over $R/N(R)$
a system $(\overline{\Sigma},\overline{\Delta})$ of endomorphisms and
$\Sigma$-derivations defined by
$\overline{\sigma}_{i}(\overline{r}):=\overline{\sigma_{i}(r)}$ and
$\overline{\delta}_{i}(\overline{r}):=\overline{\delta_{i}(r)}$, $1\leq i\leq
n$. Since $N(R)$ is proper, [22], Proposition 2.6 (ii), implies that
$A/N(R)\langle x_{1},\dots,x_{n}\rangle$ is a skew PBW extension over
$R/N(R)$. Note that $R/N(R)$ is a $\overline{\Sigma}$-rigid ring, since if
$\overline{r}\overline{\sigma}^{\alpha}(\overline{r})=0$, then
$\overline{r\sigma^{\alpha}(r)}=0$, and so $r\sigma_{i}(r)\in N(R)$. Having in
mind that $N(R)$ is $\Sigma$-rigid, $r\in N(R)$ and so $\overline{r}=0$. By
[38], Theorem 4.4, $A/N(R)\langle x_{1},\dots,x_{n}\rangle$ is a reduced ring
(cf. [7], Theorem 6.1.9).

Let us see that $N(A)\subseteq N(R)\langle x_{1},\dots,x_{n}\rangle$. If $f\in
N(A)$, then $f^{k}=0$, for some positive integer $k$. Thus
$\overline{f^{k}}=\overline{f}^{k}=0$ in $A/N(R)\langle
x_{1},\dots,x_{n}\rangle$, and so $\overline{f}\in N\left(A/N(R)\langle
x_{1},\dots,x_{n}\rangle\right)=0$. Hence $f\in N(R)\langle
x_{1},\dots,x_{n}\rangle$, that is, $N(A)\subseteq N(R)\langle
x_{1},\dots,x_{n}\rangle$.

For the another inclusion, if $f=r_{0}+r_{1}X_{1}+\cdots+r_{t}X_{t}\in
N(R)\langle x_{1},\dots,x_{n}\rangle$, where $r_{0},r_{1},\dots,r_{t}\in
N(R)\subseteq N(A)$, since $N(A)$ is an ideal of $A$, then
$r_{0},r_{1}X_{1},\dots,r_{t}X_{t}\in N(A)$, and therefore
$f=r_{0}+r_{1}X_{1}+\cdots+r_{t}X_{t}\in N(A)$.

(ii) $\Rightarrow$ (i) Let $r\in N(R)\subseteq N(A)$. Then $rx_{i}\in
N(R)\langle x_{1},\dots,x_{n}\rangle$, for $1\leq i\leq n$, whence
$x_{i}r=\sigma_{i}(r)x_{i}+\delta_{i}(r)\in N(R)\langle
x_{1},\dots,x_{n}\rangle=N(A)$. By assumption $N(R)$ is a $\Sigma$-ideal, so
$\sigma_{i}(r)\in N(R)$ and thus $\delta_{i}(r)\in N(R)$, i.e., $N(R)$ is
$\Delta$-invariant. Since $N(R)$ is proper, [22], Proposition 2.6 (i), implies
that $N(R)\langle x_{1},\dots,x_{n}\rangle$ is an ideal of $A$ and
$AN(R)\subseteq N(R)\langle x_{1},\dots,x_{n}\rangle$. Using that $N(R)\langle
x_{1},\dots,x_{n}\rangle=N(A)$, we obtain that $N(A)$ is an ideal of $A$, and
hence $A$ is NI.

To show that $N(R)$ is $\Sigma$-rigid it is enough to see that for $r\in R$
and $1\leq i\leq n$, $r\sigma_{i}(r)\in N(R)$ implies that $r\in N(R)$. If for
$r\in R$, $r\sigma_{i}(r)\in N(R)$, $1\leq i\leq n$, then
$r\sigma_{i}(r)x_{i}\in N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$, $1\leq
i\leq n$, and so
$\sigma_{i}(r)x_{i}r=\sigma_{i}(r)\sigma_{i}(r)x_{i}+\sigma_{i}(r)\delta_{i}(r)\in
N(R)\langle x_{1},\dots,x_{n}\rangle$. Thus $\sigma_{i}(r^{2})\in
N(R)=\sigma_{i}^{-1}(N(R))$, since $N(R)$ is $\Sigma$-ideal. Then $r^{2}\in
N(R)$ and so $r\in N(R)$.

(i) $\Rightarrow$ (iii) If $A$ is NI then $N^{*}(A)=N(A)$, and by Proposition
3.2, $R$ is NI, i.e., $N^{*}(R)=N(R)$. From implication (i) $\Rightarrow$ (ii)
we have that $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$, and so
$N^{*}(A)=N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle$.

(iii) $\Rightarrow$ (ii) Suppose that $N(R)$ is a $\Sigma$-rigid ideal of $R$.
By the same argument as in the proof of (i) $\Rightarrow$ (ii), we have that
$N(R)$ is a $\Sigma$-ideal of $R$. If $r\in N(R)$, then
$x_{i}r=\sigma_{i}(r)x_{i}+\delta_{i}\in AN(R)\subseteq N(R)\langle
x_{1},\dots,x_{n}\rangle$, and since $\sigma_{i}(r)\in N(R)$ then
$\delta_{i}(r)\in N(R)$, for $1\leq i\leq n$, which shows that $N(R)$ is
$\Delta$-invariant. Using the same argument as in the proof of (i)
$\Rightarrow$ (ii), we see that $N(A)\subseteq N(R)\langle
x_{1},\dots,x_{n}\rangle$. Since $N(R)$ is an ideal then $R$ is NI, and so
$N^{*}(R)=N(R)$. By assumption $N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle=N^{*}(A)$, whence $N(R)\langle
x_{1},\dots,x_{n}\rangle=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle=N^{*}(A)\subseteq N(A)$. ∎

As a particular case of Theorem 3.8, we have the following corollary.

######  Corollary 3.9 ([32], Theorem 3.1).

Let $R$ be a ring, $\sigma$ an endomorphism of $R$, and $\delta$ a
$\sigma$-derivation of $R$. Then the following statements are equivalent:

  1. (i)

$R[x;\sigma,\delta]$ is NI and $N(R)$ is $\sigma$-rigid.

  2. (ii)

$N(R)$ is a $\sigma$-ideal of $R$ and
$N(R[x;\sigma,\delta])=N(R)[x;\sigma,\delta]$.

  3. (iii)

$N(R)$ is a $\sigma$-rigid ideal of $R$ and
$N^{*}(R[x;\sigma,\delta])=N^{*}(R)[x;\sigma,\delta]$.

##  4\. NJ skew PBW extensions

In this section, we present the original results of the paper about NJ
property for skew PBW extensions.

We start with the following assertion that follows directly from definitions
of NI and NJ rings.

######  Proposition 4.1.

If $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ is an NJ skew PBW extension
over a ring $R$, then $R$ is NI and $J(A)\cap R=N(R)$.

From [7], Propositions 3.2.1 and 3.2.3, it follows the next result.

######  Proposition 4.2.

Skew PBW extensions over a domain $R$ are NJ rings, and hence, NI rings.

######  Theorem 4.3.

If $A$ is a graded skew PBW extension over
$R=\bigoplus_{n\in\mathbb{N}}R_{n}$, then $A$ is NJ if and only if $A$ is NI
and $J(A)\cap R_{0}$ is a nil ideal.

###### Proof.

Suppose that $A$ is NI and $J(A)\cap R_{0}$ is nil. By [45], Remark 2.10 (i),
$R_{0}=A_{0}$, whence $J(A)\cap R_{0}=J(A)\cap A_{0}$ is nil. From [32],
Theorem 2.4, $J(A)=N(A)$, i.e., $A$ is an NJ ring.

Conversely, if $A$ is an NJ ring, then $A$ is NI. Since $R_{0}$ is a subring,
$J(A)\cap R_{0}$ is an ideal, and as $A$ is an NJ ring, $J(A)=N(A)$. In this
way, $J(A)\cap R_{0}\subseteq J(A)=N(A)$, and hence $J(A)\cap R_{0}$ is a nil
ideal. ∎

######  Corollary 4.4.

If $A$ is a graded skew PBW extension over a connected algebra
$R=\bigoplus_{n\in\mathbb{N}}R_{n}$, then $A$ is NJ if and only if $A$ is NI.

###### Proof.

Since $R$ is connected, [45], Remark 2.10 (i), implies that
$A_{0}=R_{0}=\mathbb{K}$, whence $J(A)\cap A_{0}=\\{0\\}$ is nil. The result
follows from Theorem 4.3. ∎

######  Corollary 4.5.

Let $A$ be a quasi-commutative skew PBW extension over a ring $R$. Then $A$ is
NJ if and only if $A$ is NI and $J(A)\cap R$ is a nil ideal of $R$.

###### Proof.

From [46], Proposition 2.6, we know that quasi-commutative skew PBW extensions
where $R$ has the the trivial graduation are graded skew PBW extensions, so
[45], Remark 2.10 (i), guarantees that $A_{0}=R_{0}=R$. Thus, the assertion
follows from Theorem 4.3. ∎

######  Corollary 4.6.

If $A$ is an NJ quasi-commutative skew PBW extension over a ring $R$ and
$J(R)\subseteq J(A)$ then $R$ is NJ.

###### Proof.

By Proposition 4.1 we have that $J(A)\cap R=N(R)$. Since $J(R)\subseteq J(A)$
then $J(R)\subseteq N(R)$. Since $A$ is NJ then it is NI, so by Proposition
3.2 we have that $N(R)\subseteq J(R)$. Therefore $R$ is NJ. ∎

The next theorem presents similar results to [17], Theorem 3.10 (1).

######  Theorem 4.7.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a quasi-commutative
bijective skew PBW extension over $R$. If $R$ is a weakly 2-primal weak
$\Sigma$-compatible ring, then $A$ is NJ.

###### Proof.

Since $A$ is quasi-commutative, $\delta_{i}=0$ for $1\leq i\leq n$. Thus $A$
is weak $\Delta$-compatible. If $R$ is weakly 2-primal then it is NI, and by
Theorem 3.3, $A$ is NI. Let us show that $J(A)\cap R$ is nil. If $r\in
J(A)\cap R$, then $rx_{1}\in J(A)$. From [46], Proposition 2.6, $A$ is a
graded skew PBW extension with the trivial graduation of $R$, and so $rx_{1}$
is a homogeneous component of $J(A)$ with degree 1. By [32], Lemma 2.3 (2),
$rx_{1}\in N(A)$. In this way, [41], Theorem 4.6, implies that $r\in N(R)$.
The assertion follows from Corollary 4.5. ∎

######  Remark 4.8.

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

  1. (i)

If $R$ is 2-primal and $(\Sigma,\Delta)$-compatible, then by Proposition 3.1
(i) we have that $A$ is NI, that is, $N^{*}(A)=N(A)$. Now, from [28], Theorem
4.11, $J(A)=N_{*}(A)$, whence $N(A)=J(A)$ and thus $A$ is NJ.

  2. (ii)

If $R$ is locally finite and weak $\Sigma$-skew Armendariz, then $R$ is NJ.
More exactly, since $R$ is weak $\Sigma$-skew Armendariz, [38], Proposition
4.9, implies that $R$ is Abelian. Now, by assumption $R$ is locally finite, so
[15], Proposition 2.5, guarantees that $N(R)=J(R)$, that is, $R$ is an NJ
ring.

  3. (iii)

If $A$ is NJ and $J(R)$ is nil (or $N^{*}(R)=J(R)$ or $R/N^{*}(R)$ is
semiprimitive), then $R$ is NJ. Note that if $A$ is NJ, Proposition 4.1 shows
that $R$ is NI, and if $J(R)$ is nil (or $N^{*}(R)=J(R)$ or $R/N^{*}(R)$ is
semiprimitive), then by [17], Proposition 2.3, $R$ is NJ.

######  Proposition 4.9.

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

  1. (i)

If $A$ is NI and $R$ is $\Delta$-compatible and right duo, then $A$ is NJ.

  2. (ii)

$A$ is NI and $I\subseteq N(R)$ if and only if $A$ is NJ, where $I\subseteq R$
is the set of all coefficients of all terms of all polynomials of $J(A)$.

###### Proof.

(i) If $A$ is NI then $N(A)=N^{*}(A)\subseteq J(A)$. For the other inclusion,
if $f=a_{0}+a_{1}X_{1}+\dots+a_{t}X_{t}\in J(A)$, then
$fx_{n}=a_{0}x_{n}+a_{1}X_{1}x_{n}+\dots+a_{t}X_{t}x_{n}\in J(A)$ and
therefore $1+fx_{n}=1+a_{0}x_{n}+a_{1}X_{1}x_{n}+\dots+a_{t}X_{t}x_{n}$ is a
unit of $A$. Since $R$ is $(\Sigma,\Delta)$-compatible and right duo, [10],
Theorem 4.7, guarantees that $a_{0},a_{1},\dots,a_{t}\in N(R)$. So,
$f=a_{0}+a_{1}X_{1}+\dots+a_{t}X_{t}\in N(R)\langle x_{1},\dots,x_{n}\rangle$.
As $A$ is NI, Proposition 3.7 (iv) implies that $N(R)\langle
x_{1},\dots,x_{n}\rangle=N(A)$, that is, $f\in N(A)$.

(ii) Suppose that $A$ is NI and $I\cap R\subseteq N(R)$. By Proposition 3.7
(iv), $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$. If
$f=a_{0}+a_{1}X_{1}+\dots+a_{t}X_{t}\in J(A)$, then by hypothesis $a_{k}\in
N(R)$, for $0\leq k\leq t$, whence $a_{k}X_{k}\in N(R)\langle
x_{1},\dots,x_{n}\rangle$, for every $k$. Since $A$ is NI, then $N(A)$ is an
ideal of $A$, and hence $f=a_{0}+a_{1}X_{1}+\dots+a_{t}X_{t}\in N(A)$.

For the converse, suppose that $A$ is NJ. Then $A$ is NI, and so Proposition
3.7 (iv) implies that $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$. Now, if
$r\in I$, then for some $f=a_{0}+a_{1}X_{1}+\cdots+a_{t}X_{t}\in J(A)$ there
exists $1\leq k\leq t$ such that $r=a_{k}$. Since $J(A)=N(A)=N(R)\langle
x_{1},\dots,x_{n}\rangle$, it follows that $r\in N(R)$. ∎

The following theorem is another important result of the paper.

######  Theorem 4.10.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension of
derivation type. Then $A$ is NI if and only if $A$ is NJ.

###### Proof.

By Proposition 4.9 (ii) it is enough to prove that every coefficient of each
term of each polynomial of $J(A)$ is nilpotent. Let
$f=r_{0}+r_{1}X_{1}+\dots+r_{t}X_{t}\in J(A)$. Since $A$ is NI then $N(A)$ is
an ideal of $A$, and by Proposition 3.7 (iv), $N(R)$ is a $\Delta$-invariant
ideal of $R$ and $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$. Since $A$ is of
derivation type, $N(R)$ is a $\Sigma$-ideal. By [22], Proposition 2.6 (ii),
$A/N(R)\langle x_{1},\dots,x_{n}\rangle$ is a skew PBW extension of $R/N(R)$.
By considering the notation of the proof of [22], Proposition 2.6 (ii), and
identifying $\overline{x_{i}}$ with $x_{i}$, $1\leq i\leq n$, we use
$A/N(A)=A/N(R)\langle
x_{1},\dots,x_{n}\rangle\cong\overline{\sigma}(R/N(R))\langle
x_{1},\dots,x_{n}\rangle$ to denote such an extension. Now, by [22],
Proposition 2.2 (i), the system of endomorphisms and $\Sigma$-derivations
$(\Sigma,\Delta)$ induce over $R/N(R)$ a system
$(\overline{\Sigma},\overline{\Delta})$ of endomorphisms and
$\Sigma$-derivations defined by
$\overline{\sigma}_{i}(\overline{r}):=\overline{\sigma_{i}(r)}$ and
$\overline{\delta}_{i}(\overline{r}):=\overline{\delta_{i}(r)}$, $1\leq i\leq
n$. If
$\overline{r}\overline{\sigma}_{i}(\overline{r})=\overline{r\sigma_{i}(r)}=0$
then $r\sigma_{i}(r)\in N(R)$ and so $r\in N(R)$, since by Proposition 3.7
(i), $N(R)$ is $\Sigma$-rigid. Therefore $\overline{r}=0$ and so $R/N(R)$ is
$\overline{\Sigma}$-rigid. By [39], Theorem 3.9, $R/N(R)\langle
x_{1},\dots,x_{n}\rangle\cong A/N(A)$ is $(\Sigma,\Delta)$-compatible (see
also [7], Proposition 6.2.4). Now, as $f\in J(A)$, then
$\overline{f}:=f+N(A)\in J(A)/N(A)=J\left(A/N(A)\right)$, since by Proposition
3.2 $N(A)\subseteq J(A)$. So,
$\overline{f}x_{n}=\overline{r_{0}}x_{n}+\overline{r_{1}}X_{1}x_{n}+\dots+\overline{r_{t}}X_{t}x_{n}\in
J\left(\overline{\sigma}(R/N(R)\langle x_{1},\dots,x_{n}\rangle\right)$, with
$\overline{r_{k}}:=r_{k}+N(R)$, $1\leq k\leq n$. Having in mind that
$\overline{f}x_{n}\in J\left(\overline{\sigma}(R/N(R)\langle
x_{1},\dots,x_{n}\rangle\right)$, then
$\overline{1}+\overline{f}x_{n}=\overline{1}+\overline{r_{0}}x_{n}+\overline{r_{1}}X_{1}x_{n}+\dots+\overline{r_{t}}X_{t}x_{n}$
is a unit of $\overline{\sigma}(R/N(R)\langle x_{1},\dots,x_{n}\rangle$. Thus,
by [28], Remark 4.9 (ii), $\overline{r_{k}}\in N(R/N(R))$, $0\leq k\leq n$.
Since $R/N(R)$ is reduced we have that $\overline{r_{k}}=0$, for each $0\leq
k\leq n$. In this way, $r_{k}\in N(R)$, for each $0\leq k\leq n$. The result
follows from Proposition 4.9 (ii). ∎

######  Remark 4.11.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension of
derivation type. By Propositions 3.7 and 4.9, and Theorem 4.10, if $A$ is NI
then

$N(A)=N^{*}(A)=N(R)\langle x_{1},\dots,x_{n}\rangle=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle=J(A).$

We immediately have the following corollary.

######  Corollary 4.12 ([31], Proposition 2.8).

Let $R$ be a ring and $\delta$ a derivation of $R$. If $R[x;\delta]$ is NI,
then
$J(R[x;\delta])=N(R[x;\delta])=N(R)[x;\delta]=N^{*}(R)[x;\delta]=N^{*}(R[x;\delta])$.

The next result, Proposition 4.13, extends [32], Corollary 3.2.

######  Proposition 4.13.

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a skew PBW extension of
derivation type, then the following statements are equivalent:

  1. (i)

$A$ is NJ.

  2. (ii)

$A$ is NI.

  3. (iii)

$R$ is NI and $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$.

  4. (iv)

$R$ is NI and $N^{*}(A)=N^{*}(R)\langle x_{1},\dots,x_{n}\rangle$.

###### Proof.

(i) $\Leftrightarrow$ (ii) It follows from Theorem 4.10.

(ii) $\Rightarrow$ (iii) If $A$ is NI then by Proposition 3.2 we have that $R$
is NI. From Proposition 3.7 (iv), $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$.

(iii) $\Rightarrow$ (ii) If $R$ is NI then $N(R)$ is an ideal of $R$, and by
Proposition 3.7 (iv), $N(R)$ is $\Delta$-invariant and therefore $A$ is NI.

(iii) $\Rightarrow$ (iv) If $R$ is NI then $N(R)$ is a $\Sigma$-ideal. Theorem
3.8, part (ii) $\Rightarrow$ (iii), shows that $N^{*}(A)=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle$.

(iv) $\Rightarrow$ (iii) Since $R$ is NI, then $N(R)$ is an ideal of $R$. Now,
by Proposition 3.7 (i), $N(R)$ is $\Sigma$-rigid. Theorem 3.8, part (iii)
$\Rightarrow$ (ii), implies that $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$.
∎

The next proposition establishes similar results to [32], Corollary 3.3.

######  Proposition 4.14.

If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a quasi-commutative skew
PBW extension of $R$, then the following statements are equivalent:

  1. (i)

$A$ is NJ and $N(A)=N(R)\langle x_{1},\dots,x_{n}\rangle$.

  2. (ii)

$N(R)$ is a $\Sigma$-ideal of $R$ and $N(A)=N(R)\langle
x_{1},\dots,x_{n}\rangle$.

  3. (iii)

$A$ is NI and $N(R)$ is $\Sigma$-rigid.

  4. (iv)

$N(R)$ is $\Sigma$-rigid ideal of $R$ and $N^{*}(A)=N^{*}(R)\langle
x_{1},\dots,x_{n}\rangle$.

###### Proof.

(i) $\Rightarrow$ (ii) If $A$ is NJ we know that $A$ is NI, and therefore $R$
is NI, so $N(R)$ is an ideal of $R$. In the proof of Theorem 3.8, part (i)
$\Rightarrow$ (ii), it was already shown that $N(R)$ is a $\Sigma$-ideal.

(ii) $\Rightarrow$ (iii) Theorem 3.8, part (ii) $\Rightarrow$ (i).

(iii) $\Rightarrow$ (i) If $r\in J(A)\cap R$ then $rx_{1}\in J(A)$. From [46],
Proposition 2.6, $A$ is a graded skew PBW extension with the trivial
graduation of $R$. Thus $rx_{1}$ is a homogeneous component of $J(A)$ with
grade 1. By [32], Lemma 2.3 (2), we have that $rx_{1}\in N(A)$. Since $A$ is
NI and $N(R)$ is $\Sigma$-rigid, then Theorem 3.8, implication (i)
$\Rightarrow$ (ii), guarantees that $N(A)=N(R)\langle
x_{1},\dots,x_{n}\rangle$. Since $rx_{1}\in N(A)$, it follows that $r\in
N(R)$, and thus $J(A)\cap R$ is a nil ideal of $R$. Corollary 4.5 implies that
$A$ is NJ.

(iii) $\Leftrightarrow$ (iv) This is precisely the content of Theorem 3.8,
equivalence (ii) $\Leftrightarrow$ (iii). ∎

##  5\. Future work

In this article, we have established necessary or sufficient conditions to
ensure that skew PBW extensions are either NI or NJ rings. Now, since the
notion of weak compatibility introduced in [41] is more general than the
notion of compatibility defined in [12] and [39], a first task is to extend
the results obtained here that used the compatibility condition, now assuming
the weak compatibility condition.

On the other hand, since skew PBW extensions are part of a more general family
of noncommutative algebraic structures such as the semi-graded rings
introduced by Lezama and Latorre [25], which have been studied recently from
the point of view of noncommutative algebraic geometry (topics such as Hilbert
series and Hilbert polynomial, generalized Gelfand-Kirillov dimension, non-
commutative schemes, and Serre-Artin-Zhang-Verevkin theorem, see Lezama et
al., [21] and [24]), and having in mind that semi-graded rings are more
general than $\mathbb{N}$-graded rings (of course, we mean rings that do not
admit a non-trivial $\mathbb{N}$-graduation), the natural task is to
investigate NI and NJ properties for semi-graded rings with the aim of
extending results established by Nasr-Isfahani [32] in the context of graded
rings.

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