# On the uniform dimension and the associated primes of skew PBW extensions

Sebastián Higuera  Universidad Nacional de Colombia - Sede Bogotá Campus
Universitario<EMAIL_ADDRESS>,  María Camila Ramírez  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

###### Abstract.

In this paper, we study the uniform dimension and the associated prime ideals
of induced modules over skew PBW extensions.

###### Key words and phrases:

Induced module, associated prime, uniform dimension, skew PBW extension.

###### 2020 Mathematics Subject Classification:

16D25, 16P60, 16S36, 16S38

The authors were supported by the research fund of Faculty of Science, Code
HERMES 53880, Universidad Nacional de Colombia - Sede Bogotá, Colombia

##  1\. Introduction

Throughout the paper, every ring $R$ is associative (not necessarily
commutative) with identity unless stated otherwise. In addition, $S$ denotes
the skew polynomial ring of $R$ (also known as Ore extension)
$R[x;\sigma,\delta]$ defined by Ore [32] where $\sigma$ is an automorphism of
$R$ and $\delta$ is a $\sigma$-derivation of $R$. In the context of the skew
polynomial rings, Leroy and Matczuk [24] considered the induced modules on
these noncommutative rings: if $M_{R}$ is a right module and $S$ is a skew
polynomial ring of $R$, then $M\otimes_{R}S:=\widehat{M}_{S}$ is said to be
the induced module of $M_{R}$ [24, p. 2745]. They investigated problems
related to the uniform dimension and the associated primes of the induced
module $\widehat{M}_{S}$ by considering good polynomials.

These polynomials were used by Shock with the aim of proving that the uniform
dimensions of a ring $R$ and the polynomial ring $R[x]$ are equal [41, Theorem
2.6]. A polynomial $f(x)\in R[x]$ is called good if the annihilators of the
coefficients of $f(x)$ are equal [41, p. 252]. A polynomial
$g\in\widehat{M}_{S}$ is called good if for any $r\in R$, ${\rm deg}(gr)={\rm
deg}(g)$ provided $gr\neq 0$ [24, Definition 3.2]. A submodule $B_{S}$ of
$\widehat{M}_{S}$ is called good, if for any good polynomial $g\in B_{S}$ and
any $n\geq{\rm deg(g)}$, there exists a good polynomial of degree $n$ in $gS$
[24, Definition 4.4]. Leroy and Matczuk characterized the good polynomials of
$\widehat{M}_{S}$ and the right annihilators of generated modules by these
polynomials [24, Lemmas 3.4 and 3.7]. In addition, they described the
essential and uniform submodules of the module $\widehat{M}_{S}$ [24, Theorem
4.6], and proved that if $\widehat{M}_{S}$ is a good module, then the uniform
dimensions of $M_{R}$ and $\widehat{M}_{S}$ are the same [24, Theorem 4.9].
They showed that all associated prime ideals of the induced module
$\widehat{M}_{S}$ arise from associated primes of the module $M_{R}$ [24,
Theorem 5.7]. Continuing with the study of associated prime ideals and as a
natural generalization of these ideals, Ouyang and Birkenmeier [33] defined
the nilpotent associated primes [33, Definition 3.2] and considered the
nilpotent good polynomials as a tool to investigate these ideals [33,
Definition 3.3]. They characterized the nilpotent associated primes of skew
polynomial rings [33, Theorem 3.1].

Gallego and Lezama [15] defined the skew PBW extensions as a generalization of
the Poincaré-Birkhoff-Witt extensions introduced by Bell and Goodearl [9] and
the skew polynomial rings of injective type. Since its introduction, ring and
homological properties of skew PBW extensions have been widely studied. In the
literature, several authors have shown that the skew PBW extensions generalize
families of noncommutative algebras such as 3-dimensional skew polynomial
algebras introduced by Bell and Smith [10], ambiskew polynomial rings in the
sense of Jordan [20], solvable polynomial rings by Kandri-Rody and
Weispfenning [21], almost normalizing extensions defined by McConnell and
Robson [29], and skew bi-quadratic algebras recently introduced by Bavula [8].
For more details about skew PBW extensions and other noncommutative algebras
having PBW bases, see [1, 14, 16, 40]. Related to the good polynomials and the
associated primes, Higuera and Reyes [19] extended the notion of nilpotent
good polynomials and characterized the nilpotent associated prime ideals over
skew PBW extensions [19, Theorem 4.4]. Annin [5] considered the annihilator-
compliant polynomials to investigate associated prime ideals of the induced
module $\widehat{M}_{S}$. According to Annin, $m(x)=m_{0}+\cdots+m_{k}x^{k}\in
M[x]$ with $m_{k}\neq 0$ is called annihilator-compliant if for each $i<k$,
${\rm ann}_{R}(m_{k})\subseteq{\rm ann}_{R}(m_{i})$ [5, Definition 2.23]. Niño
et al. [31] extended this definition to study associated prime ideals of
modules over skew PBW extensions. Under certain compatibility conditions, they
characterized the associated primes of the induced module
$M\otimes_{R}A:=M\langle X\rangle_{A}$ where $A$ is a skew PBW extension over
a ring $R$ [31, Theorem 3.12].

Thinking about the above results and motivated for the development of the
theory of the uniform dimension and the associated prime ideals of polynomial
modules over noncommutative rings of polynomial type (see [5], [24], [31],
[19], and references therein), our aim in this paper is to characterize
essential modules and the uniform dimension of induced modules over skew PBW
extensions. Additionally, we study the uniform dimension of induced modules
and investigate the associated prime ideals of induced modules over families
of rings more general than skew polynomial rings.

The paper is organized as follows. In Section 2, we recall some definitions
and preliminaries about skew PBW extensions. Section 3 presents the definition
of good polynomial and original results that characterize these polynomials
(Lemmas 3.2, 3.3, 3.8, and Proposition 3.4). We also present several results
on essential modules and uniform dimension, within which we characterize the
uniform dimension of induced modules over skew PBW extensions (Lemma 3.9, and
Theorems 3.14 and 3.16). Section 4 contains results related to the
characterization of associated primes of induced modules over these extensions
(Lemma 4.1, and Theorems 4.5 and 4.7). Our results generalize those
corresponding presented by Leroy and Matczuk [24]. 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. [2, 30, 37, 38]). In this way, the
results formulated in this paper about associated prime ideals extend or
contribute to those presented by Annin [5], Brewer and Heinzer [11], Faith
[13], Leroy and Matczuk [24], Niño et al., [31], and references therein.
Finally, Section 5 illustrates the results established in Sections 3 and 4
with several noncommutative algebras that cannot be expressed as skew
polynomial rings.

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

######  Definition 2.1 ([15, Definition 1]).

Let $R$ be a ring. A ring $A$ is said to be a skew PBW extension over $R$ (the
ring of coefficients), denoted $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$,
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 every $1\leq i\leq n$ and any $r\in R\ \backslash\ \\{0\\}$, there exists
$c_{i,r}\in R\ \backslash\ \\{0\\}$ 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

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

i.e. there exist elements $r_{0}^{(i,j)},r_{1}^{(i,j)},\dotsc,r_{n}^{(i,j)}\in
R$ with

$x_{j}x_{i}-d_{i,j}x_{i}x_{j}=r_{0}^{(i,j)}+\sum_{k=1}^{n}r_{k}^{(i,j)}x_{k}$.

Since ${\rm Mon}(A)$ is a left $R$-basis of $A$, the elements $c_{i,r}$ and
$d_{i,j}$ are unique. Thus, every non-zero element $f\in A$ can be uniquely
expressed as $f=\sum_{i=0}^{m}a_{i}X_{i}$, with $a_{i}\in R$, $X_{0}=1$, and
$X_{i}\in\text{Mon}(A)$, for $0\leq i\leq m$ [15, Remark 2].

######  Proposition 2.2 ([15, Proposition 3]).

If $A$ is a skew PBW extension over $R$, then 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 $1\leq i\leq n$, where
$r\in R$.

We use the notation $\Sigma:=\\{\sigma_{1},\dots,\sigma_{n}\\}$ and
$\Delta:=\\{\delta_{1},\dots,\delta_{n}\\}$ for the families of injective
endomorphisms and derivations of Proposition 2.2, respectively.

######  Definition 2.3.

Let $A$ be a skew PBW extension over $R$.

  * (i)

[15, Definition 4] $A$ is called quasi-commutative if the conditions
${\rm(iii)}$ and ${\rm(iv)}$ presented above are replaced by the following:

    1. (iii’)

For every $1\leq i\leq n$ and $r\in R\setminus\left\\{0\right\\}$, there
exists $c_{i,r}\in R\setminus\left\\{0\right\\}$ such that
$x_{i}r=c_{i,r}x_{i}$.

    2. (iv’)

For every $1\leq i,j\leq n$, there exists $d_{i,j}\in
R\setminus\left\\{0\right\\}$ such that

$x_{j}x_{i}=d_{i,j}x_{i}x_{j}$.

  * (ii)

[15, 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 $1\leq i<j\leq n$.

  * (iii)

[2, Definition 2.3] If $\sigma_{i}$ is the identity homomorphism of $R$ for
all $1\leq i\leq n$, then we say that $A$ is a skew PBW extension of
derivation type. Similarly, if $\delta_{i}\in\Delta$ is zero, for every $1\leq
i\leq n$, then $A$ is called a skew PBW extension of endomorphism type.

######  Remark 2.4.

Some relationships between skew polynomial rings and skew PBW extensions are
the following:

  * (i)

If $A$ is a quasi-commutative skew PBW extension, then $A$ is isomorphic to an
iterated skew polynomial ring of endomorphism type [25, Theorem 2.3].

  * (ii)

In general, skew polynomial rings of injective type are strictly contained in
skew PBW extensions [25, Example 5(3)]. This fact is not possible for PBW
extensions. For instance, the quantum plane $\Bbbk\\{x,y\\}/\langle xy-qyx\mid
q\in\Bbbk^{*}\rangle$ is a skew polynomial ring of injective type given by
$\Bbbk[y][x;\sigma]$, where $\sigma(y)=qy$, but cannot be expressed as a PBW
extension.

  * (iii)

Skew PBW extensions of endomorphism type are more general than iterated skew
polynomial rings of endomorphism type [42, Remark 2.4 (ii)].

######  Definition 2.5 ([15, Section 3]).

If $A$ is a skew PBW extension over $R$, then:

  1. (i)

For any element $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$, we
will write
$\sigma^{\alpha}:=\sigma_{1}^{\alpha_{1}}\circ\dotsb\circ\sigma_{n}^{\alpha_{n}}$,
$\delta^{\alpha}=\delta_{1}^{\alpha_{1}}\circ\dotsb\circ\delta_{n}^{\alpha_{n}}$,
where $\circ$ denotes composition. 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 write
$x^{\alpha}\succ x^{\beta}$. If $f$ is a non-zero element of $A$, then we use
expressions as $f=a_{1}x^{\alpha_{1}}+\cdots+a_{k}x^{\alpha_{k}}$, with
$a_{i}\in R$, and $x^{\alpha_{k}}\succ\dotsb\succ x^{\alpha_{1}}$. With this
notation, we define ${\rm lm}(f):=x^{\alpha_{k}}$, the leading monomial of
$f$; ${\rm lc}(f):=a_{k}$, the leading coefficient of $f$; ${\rm
lt}(f):=a_{k}x^{\alpha_{k}}$, the leading term of $f$. Note that
$\deg(f):={\rm max}\\{\deg(x^{\alpha_{i}})\\}_{i=1}^{k}$. If $f=0$, ${\rm
lm}(0):=0$, ${\rm lc}(0):=0$, ${\rm lt}(0):=0$.

The next proposition is very useful when one need to make some computations
with elements of skew PBW extensions.

######  Proposition 2.6 ([15, 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 $A$ 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 $p_{\alpha,\beta}\neq 0$.

It is important to mention that the coefficients of the polynomial expressions
$p_{\alpha,r}$ and $p_{\alpha,\beta}$ in Proposition 2.6 are several
evaluations of $r$ in $\sigma$’s and $\delta$’s depending on the coordinates
of $\alpha$. A more explicit description of these relations can be found in
[35, Proposition 2.9 and Remark 2.10 (iv)].

According to Definition 2.1, if $A$ is a skew PBW extension over a ring $R$,
then $A$ is a free left $R$-module. In this way, if $M_{R}$ is a right module,
we can consider the set $M\langle X\rangle_{A}$ where the elements are of the
form $m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}},m_{i}\in M_{R}$ and
$x^{\alpha_{i}}\in{\rm Mon}(A)$, for every $1\leq i\leq k$. Considering
Proposition 2.6 and the explicit relations described in [35, Proposition 2.9
and Remark 2.10 (iv)], the set $M\langle X\rangle_{A}$ has $A$-module
structure defined as follows: if $mx^{\alpha_{i}}:=mx_{1}^{\alpha_{i1}}\cdots
x_{n}^{\alpha_{in}}\in M\langle X\rangle_{A}$ and
$bx^{\beta_{j}}:=bx_{1}^{\beta_{j1}}\cdots x_{n}^{\beta_{jn}}\in A$, then we
multiply these elements following the rule

$\begin{split}\begin{aligned} mx_{1}^{\alpha_{i1}}\cdots
x_{n}^{\alpha_{in}}bx_{1}^{\beta_{j1}}\cdots
x_{n}^{\beta_{jn}}&=m\sigma^{\alpha_{i}}(b)x^{\alpha_{i}}x^{\beta_{j}}+mp_{\alpha_{i1},\sigma_{i2}^{\alpha_{i2}}(\cdots(\sigma_{in}^{\alpha_{in}}(b)))}x_{2}^{\alpha_{i2}}\cdots
x_{n}^{\alpha_{in}}x^{\beta_{j}}\\\ &\ \ \
+mx_{1}^{\alpha_{i1}}p_{\alpha_{i2},\sigma_{i3}^{\alpha_{i3}}(\cdots(\sigma_{in}^{\alpha_{in}}(b)))}x_{3}^{\alpha_{i3}}\cdots
x_{n}^{\alpha_{in}}x^{\beta_{j}}\\\ &\ \ \
+mx_{1}^{\alpha_{i1}}x_{2}^{\alpha_{i2}}p_{\alpha_{i3},\sigma_{i4}^{\alpha_{i4}}(\cdots(\sigma_{in}^{\alpha_{in}}(b)))}x_{4}^{\alpha_{i4}}\cdots
x_{n}^{\alpha_{in}}x^{\beta_{j}}\\\ &\ \ \
+\cdots+mx_{1}^{\alpha_{i1}}x_{2}^{\alpha_{i2}}\cdots
x_{(n-2)}^{\alpha_{i(n-2)}}p_{\alpha_{i(n-1)},\sigma_{in}^{\alpha_{in}}(b)}x_{n}^{\alpha_{in}}x^{\beta_{j}}\\\
&\ \ \ +mx_{1}^{\alpha_{i1}}x_{2}^{\alpha_{i2}}\cdots
x_{(n-1)}^{\alpha_{i(n-1)}}p_{\alpha_{in},b}x^{\beta_{j}}.\end{aligned}\end{split}$

This guarantees that $M\langle X\rangle_{A}$ is a right $A$-module and is
called the induced module of $M_{R}$. If $\succeq$ is a total order defined on
${\rm Mon}(A)$ and $m\in M\langle X\rangle_{A}$ with $m\neq 0$, then we use
expressions as $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}$, where
$m_{i}\in M_{R}$, and $x^{\alpha_{k}}\succ\dotsb\succ x^{\alpha_{1}}$. With
this notation, we define ${\rm lm}(m):=x^{\alpha_{k}}$, the leading monomial
of $m$; ${\rm lc}(m):=m_{k}$, the leading coefficient of $m$; ${\rm
lt}(m):=m_{k}x^{\alpha_{k}}$, the leading term of $m$. Note that
$\deg(m):={\rm max}\\{\deg(x^{\alpha_{i}})\\}_{i=1}^{k}$. More details and
properties about the induced module $M\langle X\rangle_{A}$ can be consulted
in [31, 36].

##  3\. Good polynomials and uniform dimension

In this section, we study good polynomials, the essential modules and the
uniform dimension of induced modules over skew PBW extensions.

###  3.1. Good polynomials over skew PBW extensions

Since Leroy and Matczuk [24] used the good polynomials to study the uniform
dimension and the associated primes of induced modules over skew polynomial
rings, we introduce the following definition thinking about skew PBW
extensions.

######  Definition 3.1.

Let $A$ be a skew PBW extension over $R$, $M_{R}$ be a right module, and
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in M\langle X\rangle_{A}$
with leading coefficient $m_{k}\neq 0$. We say that $m$ is a good polynomial
if for every $r\in R$, ${\rm lm}(mr)={\rm lm}(m)$, as long as $mr\neq 0$.

Following Leroy and Matczuk, if $R$ is a ring, $\sigma$ is an automorphism of
$R$, and $M_{R}$ is a right module, then $M_{\sigma}$ denotes the
$\sigma$-twisted module defined on the same additive structure $M_{\sigma}=M$,
where the action is defined by $m\cdot r:=m\sigma(r)$, for all $r\in R$ [24,
p. 2747]. With this in mind, we consider the following definition. If
$\Sigma=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ is a finite set of automorphisms
of $R$ and $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}$, then
$M_{\sigma^{\alpha}}$ denotes the $\sigma^{\alpha}$-twisted module where the
action of $R$ over $M_{\sigma^{\alpha}}$ is given by $m\cdot
r:=m\sigma^{\alpha}(r)=m\sigma_{1}^{\alpha_{1}}\circ\dotsb\circ\sigma_{n}^{\alpha_{n}}(r)$,
for every $r\in R$. If $m\in M$, we denote by $\langle
m\rangle_{\sigma^{\alpha}}$ the $\sigma^{\alpha}$-twisted $R$-module generated
by $m$. Additionally, if $X$ is a subset of $R$ and $\alpha\in\mathbb{N}^{n}$,
then $\sigma^{-\alpha}(X)$ denotes the inverse image of $X$ under
$\sigma^{\alpha}$, that is, if $r\in\sigma^{-\alpha}(X)$, then
$\sigma^{\alpha}(r)\in X$.

######  Lemma 3.2.

Let $A$ be a bijective skew PBW extension over $R$, $M_{R}$ be a right module,
and $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in M\langle
X\rangle_{A}$ with leading coefficient $m_{k}\neq 0$. The following statements
are equivalent:

  * (1)

$m$ is a good polynomial.

  * (2)

For all $f\in mR_{R}$, ${\rm lm}(f)\succeq x^{\alpha_{k}}$.

  * (3)

For all $f\in mA_{A}$, ${\rm lm}(f)\succeq x^{\alpha_{k}}$.

  * (4)

For any $r\in R$, $m_{k}\sigma^{\alpha_{k}}(r)=0$ if and only if $mr=0$.

  * (5)

${\rm ann}_{R}(m)=\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))$.

  * (6)

${\rm ann}_{A}(m)=\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$.

  * (7)

$mA\cong\langle m_{k}\rangle_{\sigma^{\alpha_{k}}}A$ as $A$-modules.

###### Proof.

  * $(1)\Rightarrow(2)$ If $f\in mR_{R}$, then $f=mr$, for some $r\in R$. If we assume that ${\rm lm}(f)\prec x^{\alpha_{k}}$, then ${\rm lm}(mr)\prec{\rm lm}(m)$, which contradicts that $m$ is a good polynomial. Hence $x^{\alpha_{k}}\preceq{\rm lm}(f)$, for all $f\in mR_{R}$.

  * $(2)\Rightarrow(3)$ Suppose that $x^{\alpha_{k}}\preceq{\rm lm}(f)$, for all $f\in mR_{R}$. It is clear that ${\rm lm}(m)\preceq{\rm lm}(mx^{\alpha})$, for every $\alpha\in\mathbb{N}^{n}$. Thus, if $f=mg$, for some $g\in A$, then $x^{\alpha_{k}}={\rm lm}(m)\prec{\rm lm}(mg)$, whence $x^{\alpha_{k}}\preceq{\rm lm}(f)$, for all $f\in mA_{A}$.

  * $(3)\Rightarrow(4)$ Suppose that $m_{k}\sigma^{\alpha_{k}}(r)=0$. If $mr\neq 0$, then $mr$ is a non-zero element of $mA_{A}$ such that ${\rm lm}(mr)\prec{\rm lm}(m)=x^{\alpha_{k}}$, which is a contradiction. Thus, we have $mr=0$. For the other implication, if $mr=0$ then the leading coefficient of $mr$ is zero, that is, $m_{k}\sigma^{\alpha_{k}}(r)=0$ as desired.

  * $(4)\Rightarrow(5)$ If $r\in{\rm ann}_{R}(m)$, then $mr=0$. By statement $(4)$, $m_{k}\sigma^{\alpha_{k}}(r)=0$ whence $\sigma^{\alpha_{k}}(r)\in{\rm ann}(m_{k})$. Hence, $r\in\sigma^{-\alpha_{k}}({\rm ann}(m_{k}))$, and so ${\rm ann}_{R}(m)\subseteq\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))$. For the other inclusion if $r\in\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))$, then $\sigma^{\alpha_{k}}(r)\in{\rm ann}(m_{k})$ which implies that $m_{k}\sigma^{\alpha_{k}}(r)=0$, and thus $mr=0$ by hypothesis. This proves that $\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))\subseteq{\rm ann}_{R}(m)$.

  * $(5)\Rightarrow(6)$ Assume that (5) holds, and let $g=b_{1}x^{\beta_{1}}+\cdots+b_{t}x^{\beta_{t}}\in{\rm ann}_{A}(m)$. If $mg=0$, then $m_{k}\sigma^{\alpha_{k}}(b_{t})=0$, and so $b_{t}\in{\rm ann}_{R}(m)$ by hypothesis. Thus, $m\left(b_{1}x^{\beta_{1}}+\cdots+b_{t-1}x^{\beta_{t-1}}\right)=0$ whence $m_{k}\sigma^{\alpha_{k}}(b_{t-1})=0$, and so $b_{t-1}\in{\rm ann}_{R}(m)$. Continuing this process, we have $b_{i}\in\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))$, for every $1\leq i\leq t$, and so $g\in\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$. This proves the inclusion ${\rm ann}_{A}(m)\subseteq\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$. The other inclusion is clear.

  * $(6)\Rightarrow(7)$ Assume that (6) holds. If $\phi$ is the $A$-module homomorphism of $A$ over $mA$ defined by $\phi(f)=mf$ for all $f\in A$, then it is clear that the kernel of $\phi$ is ${\rm ann}_{A}(m)$, whence $mA\cong A/{\rm ann}_{A}(m)$ by the first isomorphism theorem for $A$-modules. Additionally, if ${\rm ann}_{A}(m)=\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$ by assumption, then we get the isomorphism $mA\cong A/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$. Let $\varphi$ be the map of $R/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))\times A$ over $A/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$ defined by $\varphi(\overline{r},f):=\overline{rf}$. It is not difficult to see that $\varphi$ is bilinear and due to the universal property of the tensorial product, there exists $\overline{\varphi}$ of $R/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))\otimes_{R}A$ over $A/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$ give by $\overline{\varphi}(\overline{r}\otimes f):=\overline{rf}$ with inverse $\overline{\varphi}^{-1}$ of $A/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A$ over $R/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))\otimes_{R}A$ defined by $\overline{\varphi}^{-1}(\overline{f}):=\overline{1}\otimes f$. Thus, it follows that $A/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A\cong R/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))\otimes_{R}A$. Since $R/\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))$ is isomorphic to the $R$-module $\langle m_{k}\rangle_{\sigma^{\alpha_{k}}}$, we have $mA\cong\langle m_{k}\rangle_{\sigma^{\alpha_{k}}}A$, where the isomorphism $\phi$ is defined as $\phi(mg):=m_{k}\sigma^{\alpha_{k}}(b_{1})x^{\beta_{1}}+\cdots+m_{k}\sigma^{\alpha_{k}}(b_{t})x^{\beta_{t}}$, for every $g=b_{1}x^{\beta_{1}}+\cdots+b_{t}x^{\beta_{t}}\in A$.

  * $(7)\Rightarrow(1)$ Let $\phi$ be the isomorphism defined above. If $m$ is a polynomial such that ${\rm lm}(mr)\prec{\rm lm}(m)$, for some $r\in R$, then $m_{k}\sigma^{\alpha_{k}}(r)=0$. Thus, $\phi(mr)=m_{k}\sigma^{\alpha_{k}}(r)=0$, and since $\phi$ is injective, this implies that $mr=0$. Hence, $m$ is a good polynomial.

∎

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 [41, Proposition 2.2]. Lemma 3.3 generalize
Shock’s result and characterizes the right annihilators of good polynomials
over the induced module $M\langle X\rangle_{A}$. This lemma extends [24,
Corollary 3.5].

######  Lemma 3.3.

Let $A$ be a bijective skew PBW extension over $R$ and $M_{R}$ be a right
module. If $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in M\langle
X\rangle_{A}$ with leading coefficient $m_{k}\neq 0$, then:

  * (1)

There exists $r\in R$ such that $mr$ is good polynomial.

  * (2)

If $m$ is good polynomial, we have ${\rm ann}_{A}(m)={\rm ann}_{R}(m)A$.

###### Proof.

  * (1)

Assume that the result is false and suppose that $m\in M\langle X\rangle_{A}$
is a counterexample of minimal leading monomial, that is, $mr$ is not a good
polynomial, for every $r\in R$. In particular, if $r=1$, then $m$ is not a
good polynomial, and thus there exists $r\in R$ such that ${\rm
lm}(mr)\prec{\rm lm}(m)$. If $mr\neq 0$, then by the minimality of ${\rm
lm}(m)$, there exists $c\in R$ with $mrc$ a good polynomial. However, this
contradicts the fact that $mr$ is not a good polynomial for all $r\in R$.

  * (2)

If $m$ is good polynomial, by using equivalences $(5)$ and $(6)$ of Lemma 3.2,
we have ${\rm ann}_{A}(m)=\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))A={\rm
ann}_{R}(m)A$.

∎

A submodule $N_{R}$ of $M_{R}$ is called essential if $mR\cap N\neq 0$, for
any $0\neq m\in M_{R}$. The set of all elements $m\in M_{R}$ such that ${\rm
ann}_{R}(m)$ is an essential ideal of $R$ is said to be the singular submodule
of $M_{R}$ and is denoted by $Z(M_{R})$; $M_{R}$ is called nonsingular if
$Z(M_{R})=0$ [22, Definition 3.26]. Leroy and Matczuk [24] proved that there
exist good polynomials of any degree in a submodule of a nonsingular module.
The following proposition extends [24, Proposition 3.6].

######  Proposition 3.4.

Let $A$ be a bijective skew PBW extension over $R$, $M_{R}$ be a right module,
and $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}$ be a good polynomial of
$M\langle X\rangle_{A}$ with leading coefficient $m_{k}\neq 0$. If the
submodule $m_{k}R$ of $M_{R}$ is nonsingular, then:

  * (1)

$m_{k}A$ contains a good polynomial $f$ with leading monomial $x_{i}$ for all
$i$.

  * (2)

For any $x^{\alpha_{l}}\succeq x^{\alpha_{k}}$, there exists a good polynomial
$f\in mA$ with leading monomial $x^{\alpha_{l}}$.

###### Proof.

  * (1)

Let $g\in A$ with leading term $m_{k}x_{i}$ for some $1\leq i\leq n$. Since
$m_{k}R$ is nonsingular, there exists $0\neq b\in R$ such that
$\sigma_{i}(b)R\bigcap{\rm ann}_{R}(m_{k})=0$. Note that the leading
coefficient of the polynomial $gb$ is $m_{k}\sigma_{i}(b)$. Now, consider an
element $r\in R$ such that $\sigma_{i}(r)\in{\rm
ann}_{R}(m_{k}\sigma_{i}(b))$. In this way,
$\sigma_{i}(b)\sigma_{i}(r)\in\sigma_{i}(b)R\bigcap{\rm ann}_{R}(m_{k})=0$,
and so $\sigma_{i}(br)=0$. By the injectivity of $\sigma_{i}$, if $br=0$, then
$gbr=0$. This proves that $f=gb$ is a good polynomial with leading monomial
$x_{i}$.

  * (2)

If $m$ is a good polynomial, then there exists an isomorphism of $A$-modules
$\phi$ of $mA$ over $\langle m_{k}\rangle_{\sigma^{\alpha_{k}}}A$ such that
$\phi(mg):=m_{k}\sigma^{\alpha_{k}}(b_{1})x^{\beta_{1}}+\cdots+m_{k}\sigma^{\alpha_{k}}(b_{j})x^{\beta_{j}}$,
for every $g=b_{1}x^{\beta_{1}}+\cdots+b_{j}x^{\beta_{j}}\in A$, by Lemma 3.2
(7). Furthermore, we get $\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))={\rm
ann}_{R}(m)$, by Lemma 3.2 (5), which implies that the leading monomial of
$\phi(mg)$ is $x^{\beta_{j}}$ if and only if the leading monomial of $mg$ is
$x^{\alpha_{k}+\beta_{j}}$. Therefore, $mg\in mA$ is a good polynomial if and
only if $m_{k}{\sigma^{\alpha_{k}}}(g)\in\langle
m_{k}\rangle_{\sigma^{\alpha_{k}}}A$ is a good polynomial by Lemma 3.2 (2).

It is not difficult to see that, if $m_{k}R$ is nonsingular, then $\langle
m_{k}\rangle_{\sigma^{\alpha_{k}}}$ also is. By part (1), there exists
$g^{\prime}=\phi(mg)\in\langle m_{k}\rangle_{\sigma^{\alpha_{k}}}A$ a good
polynomial such that the leading monomial of $g^{\prime}$ is $x_{1}$. Thus,
$mg^{\prime}$ is a good polynomial of $mA$ with leading monomial $x^{\beta}$
with $\beta\in\mathbb{N}^{n}$, where $\beta_{1}=\alpha_{k1}+1$ and
$\beta_{i}=\alpha_{ki}$, for all $2\leq i\leq n$. Since the leading
coefficient of $mg^{\prime}$ belongs to $\langle
m_{k}R\rangle_{\sigma^{\alpha_{k}}}$, its leading coefficient satisfies the
hypothesis of the theorem. Thus, there exists $mg^{\prime\prime}\in mA$ with
leading monomial $x^{\beta}$ where $\beta_{1}=\alpha_{k1}+2$ and
$\beta_{i}=\alpha_{ki}$, for every $2\leq i\leq n$. Following this argument,
in at most $|\alpha_{l1}-\alpha_{k1}|$ steps, we find a good polynomial
$mg_{1}$ with leading monomial $x^{\beta}$ where $\beta_{1}=\alpha_{l1}$ and
$\beta_{i}=\alpha_{ki}$, for all $2\leq i\leq n$. The idea is to continue with
$x_{2}$. By part (1), there exists $g_{1}^{\prime}\in\langle
m_{k}\rangle_{\sigma^{\alpha_{k}}}A$ a good polynomial where the leading
monomial of $g_{1}^{\prime}$ is $x_{2}$. Thus, $mg_{1}^{\prime}$ is a good
polynomial of $mA$ with leading monomial $x^{\beta}$ with
$\beta\in\mathbb{N}^{n}$, where $\beta_{1}=\alpha_{l1}$,
$\beta_{2}=\alpha_{k2}+1$ and $\beta_{i}=\alpha_{ki}$, for all $3\leq i\leq
n$. Repeating the process, in at most $|\alpha_{l2}-\alpha_{k2}|$ steps, we
find a good polynomial $mg_{2}$ with leading monomial $x^{\beta}$ where
$\beta_{i}=\alpha_{li}$ for $i=1,2$ and $\beta_{i}=\alpha_{ki}$, for all
$3\leq i\leq n$. In this way, in at most $n\cdot{\rm
max}\\{|\alpha_{li}-\alpha_{ki}|\\}$ steps, we have a good polynomial $f$ with
${\rm lm}(f)=x^{\alpha_{l}}$.

∎

Let $R$ be a ring, $\Sigma:=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ be a finite
set of endomorphisms of $R$ and $\Delta:=\\{\delta_{1},\ldots,\delta_{n}\\}$
be a finite set of $\Sigma$-derivations of $R$. If $I$ is a two-sided ideal of
$R$, then $I$ is called $\Sigma$-invariant if $\sigma_{i}(I)\subseteq I$, for
every $1\leq i\leq n$; $I$ is a $\Delta$-invariant ideal if
$\delta_{i}(I)\subseteq I$, for all $1\leq i\leq n$; if $I$ is both $\Sigma$
and $\Delta$-invariant, we say that $I$ is $(\Sigma,\Delta)$-invariant [2,
Definition 2.1]. These ideals have been widely studied in the literature [2,
26, 30, 34].

Leroy and Matczuk defined some invariant ideals associated with an ideal $I$
of $R$, an endomorphism $\sigma$ of $R$, and a $\sigma$-derivation $\delta$ of
$R$ (cf. [24, p. 2746]). Following their ideas, we consider some ideals with
the aim of studying properties of induced modules over skew PBW extensions.

######  Definition 3.5.

Let $R$ be a ring, $\Sigma:=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ be a finite
set of endomorphisms of $R$ and $\Delta:=\\{\delta_{1},\ldots,\delta_{n}\\}$
be a finite set of $\Sigma$-derivations of $R$. If $I$ is a two-sided ideal of
$R$, then

$\displaystyle I_{\Sigma}$ $\displaystyle:=\\{a\in I\ |\ \sigma^{\alpha}(a)\in
I,\ \text{for all}\ \alpha\in\mathbb{N}^{n}\\},$ $\displaystyle I_{\Delta}$
$\displaystyle:=\\{a\in I\ |\ \delta^{\beta}(a)\in I,\ \text{for all}\
\beta\in\mathbb{N}^{n}\\},$ $\displaystyle I_{\Sigma,\Delta}$
$\displaystyle:=\\{a\in I\ |\ \sigma^{\alpha}\delta^{\beta}(a)\in I,\
\text{for all}\ \alpha,\beta\in\mathbb{N}^{n}\\}.$

######  Remark 3.6.

Notice that $I$ is $\Sigma$-invariant if and only if $I=I_{\Sigma}$; $I$ is
$\Delta$-invariant if and only if $I=I_{\Delta}$, and $I$ is a
$(\Sigma,\Delta)$-invariant ideal if and only if $I=I_{\Sigma,\Delta}$.

The following lemma extends [24, Lemma 3.1].

######  Lemma 3.7.

Let $R$ be a ring, $\Sigma:=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ be a finite
set of endomorphisms of $R$, and $\Delta:=\\{\delta_{1},\ldots,\delta_{n}\\}$
be a finite set of $\Sigma$-derivations of $R$.

  * (1)

$I_{\Sigma,\Delta}\subseteq I_{\Sigma}\cap I_{\Delta}$.

  * (2)

If either $I_{\Sigma}$ is $\Delta$-invariant or $I_{\Delta}$ is
$\Sigma$-invariant, then $I_{\Sigma,\Delta}=I_{\Sigma}$.

  * (3)

If $N_{R}$ is a right module with $I={\rm ann}_{R}(N)$ an ideal
$\Delta$-invariant, then:

    * (a)

${\rm ann}_{R}(N\left\langle X\right\rangle)=I_{\Sigma,\Delta}$.

    * (b)

If ${\rm ann}_{A}(N\left\langle X\right\rangle)=JA$ for some ideal $J$, then
$J=I_{\Sigma,\Delta}={\rm ann}_{R}(N\left\langle X\right\rangle)$.

###### Proof.

  * (1)

If $a\in I_{\Sigma,\Delta}$, then $\sigma^{\alpha}\delta^{\beta}(a)\in I$ for
every $\alpha,\beta\in\mathbb{N}^{n}$. Thus, $\sigma^{\alpha}(a)\in I$ which
implies that $a\in I_{\Sigma}$. Similarly, if
$\sigma^{\alpha}\delta^{\beta}(a)\in I$ for all
$\alpha,\beta\in\mathbb{N}^{n}$, then $\delta^{\beta}(a)\in I$ for all
$\beta\in\mathbb{N}^{n}$, which implies that $a\in I_{\Delta}$, and so $a\in
I_{\Sigma}\cap I_{\Delta}$.

  * (2)

Suppose that $I_{\Sigma}$ is a $\Delta$-invariant ideal. If $a\in I_{\Sigma}$
and $I_{\Sigma}$ is $\Delta$-invariant, then $\delta^{\beta}(a)\in
I_{\Sigma}$, and hence $\sigma^{\alpha}\delta^{\beta}(a)\in I$, for all
$\alpha,\beta\in\mathbb{N}^{n}$. Thus, $I_{\Sigma}\subseteq
I_{\Sigma,\Delta}$. If $I_{\Delta}$ is $\Sigma$-invariant, then the argument
is similar.

  * (3)

Let $N_{R}$ be a right module with $I={\rm ann}_{R}(N)$ an ideal
$\Delta$-invariant.

    * (a)

Let $n=n_{1}x^{\alpha_{1}}+\cdots+n_{k}x^{\alpha_{k}}\in N\langle
X\rangle_{A}$. For every $r\in R$, the coefficients of $nr$ are products of
$n_{i}$ with elements obtained evaluating $\sigma$’s and $\delta$’s in the
element $r$ [35, Remark 2.10]. If $r\in I_{\Sigma,\Delta}$, then $nr=0$, and
so $I_{\Sigma,\Delta}\subseteq{\rm ann}_{R}(N\langle X\rangle)$. For the other
inclusion, if $r\in{\rm ann}_{R}(N\langle X\rangle)$, then $nx^{\alpha}r=0$,
for any $\alpha\in\mathbb{N}^{n}$ and every $n\in N$. Notice that the leading
coefficient of $nx^{\alpha}r$ is zero, that is, $n\sigma^{\alpha}(r)=0$, for
all $\alpha$, which implies that $r\in I_{\Sigma}$. Since $I$ is a
$\Delta$-invariant ideal, we have $r\in I_{\Sigma}=I_{\Sigma,\Delta}$ and so
${\rm ann}_{R}(N\langle X\rangle)\subseteq I_{\Sigma,\Delta}$.

    * (b)

By item (a), we obtain $I_{\Sigma,\Delta}\subseteq J$. For the other
inclusion, if we have the equality ${\rm ann}_{A}(N\langle X\rangle)=JA$, for
some ideal $J$, then $nx^{\alpha}rx^{\beta}=0$, for every
$\alpha,\beta\in\mathbb{N}^{n}$, $r\in J$, and $n\in N$. In particular, if
$\beta=0$, then $nx^{\alpha}r=0$ and $n\sigma^{\alpha}(r)=0$, for any
$\alpha$, which implies that $r\in I_{\Sigma}$. Since $I$ is
$\Delta$-invariant, we get $r\in I_{\Sigma}=I_{\Sigma,\Delta}$ which shows
that the equality $J=I_{\Sigma,\Delta}={\rm ann}_{R}(N\left\langle
X\right\rangle)$ follows.

∎

The following lemma characterizes annihilators of generated submodules by good
polynomials and extends [24, Lemma 3.7].

######  Lemma 3.8.

Let $A$ be a bijective skew PBW extension over $R$, $M_{R}$ be a right module,
and $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in M\langle
X\rangle_{A}$ be a good polynomial with leading coefficient $m_{k}\neq 0$. If
$I:=\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}R))$, then:

  * (1)

${\rm ann}_{R}(mA)\subseteq{\rm ann}_{A}(mA)\subseteq{\rm ann}_{A}(mR)$.

  * (2)

${\rm ann}_{R}(mR)=I$ and ${\rm ann}_{A}(mR)=IA$.

  * (3)

${\rm ann}_{R}(mA)=I_{\Sigma,\Delta}$.

  * (4)

If $I_{\Sigma}$ is $\Delta$-invariant, then ${\rm ann}_{A}(mA)=I_{\Sigma}A$.

###### Proof.

  * (1)

Since $R\subseteq A$, it follows that ${\rm ann}_{R}(mA)\subseteq{\rm
ann}_{A}(mA)$. On the other hand if $f\in{\rm ann}_{A}(mA)$, then $gAf=0$
whence $gRf=0$. This proves that ${\rm ann}_{A}(mA)\subseteq{\rm
ann}_{A}(mR)$.

  * (2)

Let us show that ${\rm ann}_{R}(mR)=I$. If $r\in I$, then
$m_{k}R\sigma^{\alpha_{k}}(r)=0$, and since $m$ is a good polynomial, we have
$mRr=0$ and thus $r\in{\rm ann}_{R}(mR)$. Now, if $r\in{\rm ann}_{R}(mR)$,
then $m_{k}R\sigma^{\alpha_{k}}(r)=0$ and so $r\in\sigma^{-\alpha_{k}}({\rm
ann}_{R}(m_{k}R))=I$. Let us prove that ${\rm ann}_{A}(mR)=IA$. If
$f=a_{1}x^{\beta_{1}}+\cdots+a_{t}x^{\beta_{t}}\in{\rm ann}_{A}(mR)$, then
$mRf=0$ whence $m_{k}R\sigma^{\alpha_{k}}(a_{t})=0$. Since $m$ is good, it
follows that $a_{t}\in{\rm ann}_{R}(mR)=I$. If $mRa_{t}=0$, then
$mR(a_{1}x^{\beta_{1}}+\cdots+a_{t-1}x^{\beta_{t-1}})=0$ which implies that
$m_{k}R\sigma^{\alpha_{k}}(a_{t-1})=0$. If $m$ is good, then $mRa_{t-1}=0$,
and thus $a_{t-1}\in{\rm ann}_{R}(mR)=I$. Continuing this argument, we have
that $a_{i}\in{\rm ann}_{R}(mR)=I$, for every $1\leq i\leq t$, and so
$f\in{\rm ann}_{R}(mR)A=IA$. Hence, ${\rm ann}_{A}(mR)\subseteq IA$. The
reverse inclusion is clear.

  * (3)

We have the $R$-module isomorphism $(mR)\langle X\rangle=mR\otimes_{R}A\cong
mA$. By item (2), we obtain that ${\rm ann}_{R}(mR)=I$ and by Lemma 3.7
(3)(a), it follows that ${\rm ann}_{R}((mR)\langle X\rangle)={\rm
ann}_{R}(mA)=I_{\Sigma,\Delta}$.

  * (4)

Suppose that $I_{\Sigma}$ is $\Delta$-invariant. By Lemma 3.7 (2), we get
$I_{\Sigma}=I_{\Sigma,\Delta}$. By items $(1)$ and $(3)$, we obtain
$I_{\Sigma}A={\rm ann}_{R}(mA)A\subseteq{\rm ann}_{A}(mA)$. To see the other
inclusion if $f=a_{1}x^{\beta_{1}}+\cdots+a_{t}x^{\beta_{t}}\in{\rm
ann}_{A}(mA)$, then $mRx^{\beta}f=0$ whence
$\sigma^{\beta}(a_{t})\in\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}R))=I$, for
every $\beta\in\mathbb{N}^{n}$, and so $a_{t}\in I_{\Sigma}$. Since $m$ is
good, we get $mRx^{\beta}a_{t}=0$ for all $\beta\in\mathbb{N}^{n}$ whence
$mAa_{t}=0$. Then $mA(f-a_{t}x^{\beta_{t}})=0$ and ${\rm
lm}(f-a_{t}x^{\beta_{t}})\prec{\rm lm}(f)$. An inductive argument proves that
${\rm ann}_{A}(mA)\subseteq I_{\Sigma}A$ and so the equality ${\rm
ann}_{A}(mA)=I_{\Sigma}A$ holds.

∎

###  3.2. Uniform dimension over skew PBW extensions

In this section, we study the essential modules and the uniform dimension of
induced modules over skew PBW extensions. The following theorem extends [24,
Lemma 4.1].

######  Lemma 3.9.

Let $A$ be a bijective skew PBW extension over $R$ and $M_{R}$ be a right
module. If $N_{R}$ is a submodule $M_{R}$, then:

  * (1)

$N\langle X\rangle_{R}$ is essential in $M\langle X\rangle_{R}$ if and only if
$N\langle X\rangle_{A}$ is essential in $M\langle X\rangle_{A}$.

  * (2)

${Z}(M_{R})=0$ if and only if ${Z}(M\langle X\rangle_{R})=0$.

  * (3)

If ${Z}(M_{R})=0$, then ${Z}(M\langle X\rangle_{A})=0$.

###### Proof.

  * (1)

If $N\langle X\rangle_{R}$ is essential, then $0\neq mR\cap N\langle
X\rangle_{R}\subseteq mA\cap N\langle X\rangle_{A}$ for every $m\in M\langle
X\rangle_{A}$, and so $N\langle X\rangle_{A}$ is an essential submodule of
$M\langle X\rangle_{A}$. Suppose that $N\langle X\rangle_{A}$ is an essential
submodule of $M\langle X\rangle_{A}$ and consider the $A$-module homomorphism
of $M\langle X\rangle_{A}$ over $(M/N)\langle X\rangle_{A}$ defined by
$\phi(m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}):=\overline{m_{1}}x^{\alpha_{1}}+\cdots+\overline{m_{k}}x^{\alpha_{k}}$.
It is not hard to see that $\phi$ induces an isomorphism $\overline{\phi}$
between the $A$-modules $M\langle X\rangle_{A}/N\langle X\rangle_{A}$ and
$(M/N)\langle X\rangle_{A}$. Let $\overline{f}\in(M/N)\langle X\rangle_{A}$ be
the image of $f$ by $\overline{\phi}$, for some $f\in M\langle
X\rangle_{A}\setminus N\langle X\rangle_{A}$. By Lemma 3.3 (1), there exists
$r\in R$ such that $\overline{f}r$ is a good polynomial and since $N\langle
X\rangle_{A}$ is an essential submodule of $M\langle X\rangle_{A}$, we have
$frA\cap N\langle X\rangle_{A}\neq 0$, that is, there exists
$g=b_{1}x^{\beta_{1}}+\cdots+b_{t}x^{\beta_{t}}\in A$ such that $0\neq frg\in
N\langle X\rangle_{A}$. By Lemma 3.3 (2), if $0\neq frg\in N\langle
X\rangle_{A}$, then $g\in{\rm ann}_{A}(\overline{f}r)={\rm
ann}_{R}(\overline{f}r)A$ whence $b_{i}\in{\rm ann}_{R}(\overline{f}r)$, for
every $1\leq i\leq t$. Thus, $\overline{f}rb_{i}=0$, and so $frb_{i}\in
N\langle X\rangle_{R}$. Additionally if $frg\neq 0$, then $0\neq frb_{i}\in
N\langle X\rangle_{R}$, for some $1\leq i\leq t$. This shows that $fR\cap
N\langle X\rangle_{R}\neq 0$, and hence $N\langle X\rangle_{R}$ is an
essential submodule of $M\langle X\rangle_{R}$.

  * (2)

Since $Z(M_{R})=Z(M\langle X\rangle_{R})\cap M_{R}$ if $Z(M\langle
X\rangle_{R})=0$, then $Z(M_{R})=0$. Assume that $Z(M\langle X\rangle_{R})\neq
0$ and let $f=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in Z(M\langle
X\rangle_{R})$ with leading coefficient $m_{k}\neq 0$ and minimal monomial
leading, that is, ${\rm lm}(f)\preceq{\rm lm}(g)$, for all $g\in fR$. By Lemma
3.2 (2) and (5), $f$ is a good polynomial and ${\rm
ann}_{R}(f)=\sigma^{-\alpha_{k}}({\rm ann}_{R}(m_{k}))$ is an essential ideal
of $R$ which implies that $0\neq m_{k}\in Z(M\langle X\rangle_{R})\cap
M_{R}=Z(M_{R})$.

  * (3)

Assume that $Z(M_{R})=0$ and $Z(M\langle X\rangle_{A})\neq 0$. Let
$f=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}$ be an element of
$Z(M\langle X\rangle_{A})$ with leading coefficient $m_{k}\neq 0$ and minimal
monomial leading, that is, ${\rm lm}(f)\preceq{\rm lm}(g)$, for all $g\in fA$.
By Lemma 3.2 (3), $f$ is a good polynomial and by Lemma 3.3 (2), ${\rm
ann}_{A}(f)=\sigma^{-\alpha_{k}}(I)A$ where $I={\rm ann}_{R}(a_{k})$. Since
$\sigma^{-\alpha_{k}}(I)A\cap m^{\prime}A\neq 0$, for all $m^{\prime}\in
M\langle X\rangle_{A}$, it follows that $\sigma^{-\alpha_{k}}(I)\cap
m^{\prime}M_{R}\neq 0$, and so $\sigma^{-\alpha_{k}}(I)$ is an essential ideal
of $R$. Therefore, $0\neq a_{k}\in Z(M_{R})$ which is a contradiction. This
proves that $Z(M\langle X\rangle_{A})=0$ as desired.

∎

If $R$ is a ring, $\sigma$ is an automorphism of $R$, and $\delta$ is a
$\sigma$-derivation of $R$, then an ideal $I$ of $R$ is said to be
$(\sigma,\sigma^{-1},\delta)$-stable if $\sigma(I)=I$ and $\delta(I)\subseteq
I$ [17, p. 6]. Annin [4] investigated properties of induced modules over skew
polynomial rings under the assumption that for any $m\in M_{R}$, $I={\rm
ann}_{R}(m)$ is a $(\sigma,\sigma^{-1},\delta)$-stable ideal. Leroy and
Matczuk [24] studied these same modules using the following condition: $M_{R}$
satisfies the weak $(\sigma,\delta)$-compatibility condition if every non-zero
submodule $N_{R}$ of $M_{R}$ contains an element $m\neq 0$ such that ${\rm
ann}_{R}(m)$ is $(\sigma,\sigma^{-1},\delta)$-stable [24, Definition 4.2].
Thinking about skew PBW extensions, we consider the following definition: if
$R$ is a ring, $\Sigma=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ is a finite family
of endomorphisms of $R$ and $\Delta=\\{\delta_{1},\ldots,\delta_{n}\\}$ is a
finite family of $\Sigma$-derivations of $R$, then an ideal $I$ of $R$ is
called $(\Sigma,\Sigma^{-1},\Delta)$-stable if $\sigma_{i}(I)=I$ and
$\delta_{i}(I)\subseteq I$, for every $1\leq i\leq n$. This definition allows
us to extend the compatibility condition as follows.

######  Definition 3.10.

Let $R$ be a ring, $\Sigma$ be a finite family of endomorphisms of $R$, and
$\Delta$ a finite family of $\Sigma$-derivations of $R$. We say that a right
module $M_{R}$ satisfies the weak $(\Sigma,\Delta)$-compatibility condition,
if every submodule $N_{R}$ of $M_{R}$ contains an element $a\neq 0$ such that
$I:={\rm ann}_{R}(a)$ is $(\Sigma,\Sigma^{-1},\Delta)$-stable.

The following lemma is analogous to [24, Lemma 4.3].

######  Lemma 3.11.

Let $R$ be a ring, $\Gamma=\\{\gamma_{i},\ldots,\gamma_{n}\\}$ be a family of
automorphisms of $R$, and $\Lambda=\\{\lambda_{i},\ldots,\lambda_{n}\\}$ be a
family of $\Gamma$-derivations of $R$. If $A$ is a bijective skew PBW
extension over $R$ and $M_{R}$ is a right weak $(\Gamma,\Lambda)$-compatible
module, then:

  * (1)

For any good polynomial $m\in M\langle X\rangle_{A}$, there exists $r\in R$
such that $mr$ is good and the annihilator of its leading coefficient is
$(\Gamma,\Gamma^{-1},\Lambda)$-stable.

  * (2)

Suppose that for the family of endomorphisms
$\Sigma=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ we have
$\sigma_{i}\gamma_{j}=\gamma_{j}\sigma_{i}$ for $1\leq i,j\leq n$. If there
exist some central invertible elements $q_{ij}\in R$ such that
$\sigma_{i}\lambda_{j}=q_{ij}\lambda_{j}\sigma_{i}$ and $\gamma_{i}$ and
$\lambda_{i}$ can be extended to $A$, then then module $M\langle X\rangle_{A}$
satisfies the weak $(\Gamma,\Lambda)$-compatibility condition.

###### Proof.

  * (1)

Let $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}$ be a non-zero good
polynomial with leading coefficient $m_{k}\neq 0$. If $M_{R}$ satisfies the
weak $(\Gamma,\Lambda)$-compatibility condition, then there exists $r\in R$
such that $m_{k}r\neq 0$ and $I={\rm ann}_{R}(m_{k}r)$ is
$(\Gamma,\Gamma^{-1},\Lambda)$-stable. By Lemma 3.2,
$m\sigma^{-\alpha_{k}}(r)$ is a good polynomial with leading coefficient
$m_{k}r$ and $I$ is the annihilator of its leading coefficient.

  * (2)

Let $N\langle X\rangle$ be a submodule of $M\langle X\rangle_{A}$ and
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in N\langle X\rangle$ be a
polynomial of minimal leading monomial in $mA$, that is,
$x^{\alpha_{k}}\prec{\rm lm}(f)$ for every $f\in mA$. By Lemma 3.2 and item
(1), $m$ is a good polynomial, and so $m\sigma^{-\alpha_{k}}(r)$ is also a
good polynomial for some $r\in R$ with $I={\rm ann}_{R}(m_{k}r)$
$(\Gamma,\Gamma^{-1},\Lambda)$-stable. Additionally, ${\rm
ann}_{A}(m)=\sigma^{-\alpha_{k}}(I)A$ by Lemma 3.2, and since
$\sigma_{i}\gamma_{j}=\gamma_{j}\sigma_{i}$,
$\sigma_{i}\lambda_{j}=q_{ij}\lambda_{j}\sigma_{i}$ for every $1\leq i,j\leq
n$ and some central invertible element $q_{ij}\in R$, and also $\lambda_{i}$
can be extended to $A$, it follows that ${\rm
ann}_{A}(m)=\sigma^{-\alpha_{k}}(I)A$ is $(\Gamma,\Gamma^{-1}\Lambda)$-stable.

∎

Leroy and Matczuk showed that the induced modules over skew polynomial rings
have good polynomials of any degree [24, Example 3.3]. This fact motivated the
following definition: a submodule $B_{S}$ of $\widehat{M}_{S}$ is called good,
if for any good polynomial $g\in B_{S}$ and any $n\geq{\rm deg(g)}$, there
exists a good polynomial of degree $n$ in $gS$ [24, Definition 4.4]. Following
this idea and with the purpose of studying properties of induced modules over
skew PBW extensions, we introduce the following definition.

######  Definition 3.12.

Let $A$ be a skew PBW extension over $R$ and $M_{R}$ be a right module. A
submodule $N\langle X\rangle_{A}$ of $M\langle X\rangle_{A}$ is called good,
if for any good polynomial
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in N\langle X\rangle_{A}$
and any monomial $x^{\beta}$ with $\beta\in\mathbb{N}^{n}$, there exists a
good polynomial $f\in mA$ such that ${\rm lm}(f)=x^{\beta}\succeq
x^{\alpha_{k}}$.

The following lemma generalizes [24, Lemma 4.5].

######  Lemma 3.13.

Let $A$ be a bijective skew PBW extension over $R$ and $M_{R}$ be a right
module. If one of the following conditions is satisfied

  * (1)

$M_{R}$ is nonsingular.

  * (2)

$M_{R}=R_{R}$ and for any non-zero element $r\in R$, there is a good
polynomial $m\in rA$ with leading monomial $x_{i}$ for all $i$.

  * (3)

$A$ is a skew PBW extension of endomorphism type.

  * (4)

$M_{R}$ satisfies the weak $(\Sigma,\Delta)$-compatibility condition.

then ${M}\langle X\rangle_{A}$ is a good module.

###### Proof.

  * (1)

This statement is a direct consequence of Proposition 3.4.

  * (2)

If $f=a_{1}x^{\alpha_{1}}+\cdots+a_{k}x^{\alpha_{k}}$ is a good polynomial of
$A$ with leading monomial $x^{\alpha_{k}}$, then there exists an $A$-module
isomorphism $\phi$ of $fA$ over $\langle a_{k}\rangle_{\sigma^{\alpha_{k}}}A$
such that
$\phi(fg):=a_{k}\sigma^{\alpha_{k}}(b_{1})x^{\beta_{1}}+\cdots+a_{k}\sigma^{\alpha_{k}}(b_{j})x^{\beta_{j}}$,
for every $g\in A$ with leading monomial $x^{\beta_{j}}$ for any
$\beta_{j}\in\mathbb{N}^{n}$, by Lemma 3.2 (7). Furthermore, we have
$\sigma^{-\alpha_{k}}({\rm ann}_{R}(a_{k}))={\rm ann}_{R}(f)$, by Lemma 3.2
(5), which implies that the leading monomial of $\phi(fg)$ is $x^{\beta_{j}}$
if and only if the leading monomial of $fg$ is $x^{\alpha_{k}+\beta_{j}}$.
Therefore, $fg\in fA$ is a good polynomial if and only if
$a_{k}{\sigma^{\alpha_{k}}}(g)\in\langle a_{k}\rangle_{\sigma^{\alpha_{k}}}A$
is a good polynomial by Lemma 3.2 (2).

If $x^{\beta}$ is a monomial such that $x^{\alpha_{k}}\preceq x^{\beta}$, for
some $\beta\in\mathbb{N}^{n}$, we must find a good polynomial with leading
monomial $x^{\beta}$. By assumption, if $a_{k}\in R$, there exists
$g^{\prime}=\phi(fg)\in\langle a_{k}\rangle_{\sigma^{\alpha_{k}}}A$ a good
polynomial such that the leading monomial of $g^{\prime}$ is $x_{i}$, for all
$i$. Following the same argument of Proposition 3.4, we find a good polynomial
$\overline{f}$ of $fA$ with leading monomial $x^{\beta}$ in at most
$n\cdot{\rm max}\\{|\beta_{i}-\alpha_{ki}|\\}$ steps, proving that $A_{A}$ is
a good module.

  * (3)

Let $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in M\langle
X\rangle_{A}$ be a good polynomial with leading coefficient $m_{k}\neq 0$. If
${\rm lm}(mx^{\alpha}r)\prec{\rm lm}(mx^{\alpha})$ for some $r\in R$, then
$m_{k}\sigma^{\alpha_{k}}(\sigma^{\alpha}(r))=0$. Thus, ${\rm
lm}(m\sigma^{\alpha}(r))\prec{\rm lm}(m)$, which contradicts that $m$ is good
polynomial. Hence, $mx^{\alpha}$ is also good polynomial for any
$\alpha\in\mathbb{N}^{n}$.

  * (4)

Let $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in M\langle
X\rangle_{A}$ be a good polynomial with leading coefficient $m_{k}\neq 0$. We
claim that $mx^{\beta}$ is good, for any $\beta\in\mathbb{N}^{n}$. If
$x^{\alpha_{k}+\beta}$ is the leading monomial of $mx^{\beta}$ and ${\rm
lm}(mx^{\beta}r)\prec{\rm lm}(mx^{\beta})$, for some $r\in R$, then
$m_{k}\sigma^{\alpha_{k}+\beta}(r)=0$. By Lemma 3.11 (1), ${\rm
ann}_{R}(m_{k})$ is $(\Sigma,\Sigma^{-1},\Delta)$-stable, and so $r\in{\rm
ann}_{R}(m_{k})$. Furthermore, if ${\rm ann}_{R}(m_{k})$ is
$(\Sigma,\Sigma^{-1},\Delta)$-stable, then $mx^{\beta}r=0$ which proves that
$mx^{\beta}$ is a good polynomial for any $\beta\in\mathbb{N}^{n}$.

∎

Recall that $M_{R}$ is said to be a uniform module if every submodule of
$N_{R}$ of $M_{R}$ is an essential submodule. Equivalently, $M_{R}$ is called
uniform if the intersection of any two non-zero submodules of $M_{R}$ is non-
zero [22, p. 84]. Theorem 3.14 characterizes the essential and uniform
submodules of the induced module ${M}\langle X\rangle_{A}$.

######  Theorem 3.14.

Let $A$ be bijective a skew PBW extension over $R$, $M_{R}$ be a right module,
and $N_{R}$ be a submodule of $M_{R}$. If ${N}\langle X\rangle_{A}$ is a good
module, then:

  * (1)

$N_{R}$ is essential in $M_{R}$ if and only if ${N}\langle X\rangle_{A}$ is
essential in ${M}\langle X\rangle_{A}$.

  * (2)

$N_{R}$ is uniform if and only if ${N}\langle X\rangle_{A}$ is uniform.

###### Proof.

  * (1)

If $T_{R}$ is a submodule of $M_{R}$ such that $T_{R}\cap N_{R}=0$, we have
that $T\langle X\rangle_{A}\cap N\langle X\rangle_{A}=0$, which proves that if
$N\langle X\rangle_{A}$ is essential, then $N_{R}$ is essential. Suppose that
$N_{R}$ is an essential submodule of $M_{R}$. By Lemma 3.9 (1), we need to
show that $N\langle X\rangle_{R}$ is an essential submodule in $M\langle
X\rangle_{R}$, that is, $mR\cap N\langle X\rangle_{R}\neq 0$, for any $m\in
M\langle X\rangle_{R}$. We proceed by induction on the monomials. By Lemma
3.3, we may assume that $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in
M\langle X\rangle_{R}$ is a good polynomial. If $\alpha_{k}=0$, then
$m=m_{k}\in M_{R}$, and $mR\cap N_{R}\neq 0$ because $N_{R}$ is essential.
Suppose the statement is true for any leading monomial $x^{\beta}$ such that
$x^{\beta}\prec x^{\alpha_{k}}$ with $\beta\in\mathbb{N}^{n}$. If $N_{R}$ is
an essential submodule in $M_{R}$, then $m_{k}R\cap N_{R}\neq 0$, whence
$m_{k}\sigma^{\alpha_{k}}(r)\in N_{R}$, for some $r\in R$. The element
$m_{k}\sigma^{\alpha_{k}}(r)\in N\subseteq N\langle X\rangle_{A}$ is a good
polynomial, and since $N\langle X\rangle_{A}$ is a good module, there exists a
good polynomial $g\in m_{k}\sigma^{\alpha_{k}}(r)A\subseteq N\langle
X\rangle_{A}$ such that ${\rm lm}(g)={\rm lm}(mr)$. The leading coefficient of
$g$ belongs to $m_{k}\sigma^{\alpha_{k}}(r)R$, so there exists $w\in R$ such
that $mrw$ and $g$ have the same leading coefficient, and then $g$ and $mrw$
have the same leading term. If $g=mrw$, it follows that $g\in mR\cap N\langle
X\rangle_{R}$ which proves that $N\langle X\rangle_{R}$ is essential in
$M\langle X\rangle_{R}$. If $g\neq mrw$, then $mrw-g\neq 0$ and since $mrw$
and $g$ have the same leading term, the leading monomial of $mrw-g$ is
$x^{\beta}$ for some $\beta\in\mathbb{N}^{n}$ with $x^{\beta}\prec
x^{\alpha_{k}}$. Thus, by the inductive hypothesis, there exists $s\in R$ such
that $h:=(mrw-g)s\in N\langle X\rangle_{R}$ with $h\neq 0$. Since $mrw$ and
$g$ are good polynomials of the same leading term, they have the same
annihilator in $R$. In this way, if $mrws=0$, then $gs=0$ and so $h=0$ which
is a contradiction. Therefore, we have $mrws\neq 0$, and $0\neq mrws=gs+h\in
mR\cap N\langle X\rangle_{R}$.

  * (2)

It is clear that if $N\langle X\rangle_{A}$ is a uniform module, then $N_{R}$
is a uniform module. For the other implication, suppose that $N_{R}$ is a
uniform module. If $N\langle X\rangle_{A}$ is not uniform, then there exist
non-zero polynomials $f,g\in N\langle X\rangle_{R}$ such that $fA\cap gA=0$
with $f=n_{1}x^{\alpha_{1}}+\cdots+n_{k}x^{\alpha_{k}}$ and
$g=n_{1}^{\prime}x^{\beta_{1}}+\cdots+n_{l}^{\prime}x^{\beta_{l}}$. By Lemma
3.3 (1), we may assume that $f$ and $g$ are good polynomials with ${\rm
lm}(g)\preceq{\rm lm}(f)$. Since $N\langle X\rangle_{A}$ is a good submodule
of $M\langle X\rangle_{A}$, there is a good polynomial $h\in gA$ with ${\rm
lm}(h)={\rm lm}(f)$. Let $n_{k}$ and $n_{t}^{\prime\prime}$ be the leading
coefficients of the polynomials $f$ and $h$, respectively. Since
$n_{k},n_{t}^{\prime\prime}\in N_{R}$ and $N_{R}$ is uniform, there exist
$r,s\in R$ such that $n_{k}r=n_{t}^{\prime\prime}s\neq 0$. Consider the
polynomial $z=f\sigma^{-\alpha_{k}}(r)-h\sigma^{-\alpha_{k}}(s)\in N\langle
X\rangle_{A}$ with leading monomial $x^{\beta}$ for some $\beta\in\mathbb{N}$
where $x^{\beta}\prec x^{\alpha_{k}}$. If
$f\sigma^{-\alpha_{k}}(r)=h\sigma^{-\alpha_{k}}(s)\in gA$, then
$f\sigma^{-\alpha_{k}}(r)=0$ since $fA\cap gA=0$. Thus,
$f\sigma^{-\alpha_{k}}(r)\neq h\sigma^{-\alpha_{k}}(s)$ whence $z\neq 0$.
Since $f$ and $g$ are good polynomials with ${\rm lm}(g)\preceq{\rm lm}(f)$,
$zA\cap gA\neq 0$. So, there are $v_{1},v_{2}\in A$ such that
$gv_{2}=zv_{1}=f\sigma^{-\alpha_{k}}(r)v_{1}-h\sigma^{-\alpha_{k}}(s)v_{1}$.
Since $f\sigma^{-\alpha_{k}}(r)$ and $h\sigma^{-\alpha_{k}}(s)$ are good
polynomials of the same leading term and
$f\sigma^{-\alpha_{k}}(r)v_{1}=gv_{2}+h\sigma^{-\alpha_{k}}(s)v_{1}\in fA\cap
gA=0$, then they have the same annihilators in $A$, and hence
$h\sigma^{-\alpha_{k}}(s)v_{1}=0$ which is a contradiction. Therefore,
$N\langle X\rangle_{A}$ is a uniform submodule of $M\langle X\rangle_{A}$.

∎

######  Corollary 3.15 ([24, Theorem 4.6]).

Let $S:=R[x;\sigma,\delta]$ and $N_{R}$ be a submodule of $M_{R}$ such that
$\widehat{N}_{S}$ is good. Then:

  * (1)

$N_{R}$ is essential in $M_{R}$ if and only if $\widehat{N}_{S}$ is essential
in $\widehat{M}_{S}$.

  * (2)

$N_{R}$ is uniform if and only if $\widehat{N}_{S}$ is uniform.

We recall that a module $M_{R}$ has finite uniform dimension if there exist
uniform submodules $U_{1},\ldots,U_{n}$ of $M_{R}$ such that
$U_{1}\oplus\cdots\oplus U_{n}$ is an essential submodule of $M_{R}$ [22,
Definition 6.2]. In this case, the uniform dimension of $M_{R}$ is denoted by
${\rm udim}(M_{R})=n<\infty$. It is not difficult to see that a right module
$M_{R}$ has infinite uniform dimension if and only if $M_{R}$ contains an
infinite direct sum of non-zero submodules [22, Proposition 6.4].

Theorem 3.16 establishes sufficient conditions to guarantee that $M_{R}$ and
${M}\langle X\rangle_{A}$ have the same uniform dimension (c.f. [34,
Proposition 4.10]).

######  Theorem 3.16.

Let $A$ be a bijective skew PBW extension over $R$ and $M_{R}$ be a right
module. If ${M}\langle X\rangle_{A}$ is a good module, then ${\rm
udim}({M}\langle X\rangle_{A})={\rm udim}(M_{R})$.

###### Proof.

Assume that ${\rm udim}(M_{R})=k$. There exist $N_{1},\ldots,N_{k}$ uniform
submodules of $M_{R}$ such that $N_{1}\oplus\cdots\oplus N_{k}$ is an
essential submodule of $M_{R}$. By Theorem 3.14, we have $N_{1}\langle
X\rangle_{A},\ldots,N_{k}\langle X\rangle_{A}$ are uniform submodules of
$M\langle X\rangle_{A}$, and the submodule $N_{1}\langle
X\rangle_{A}\oplus\cdots\oplus N_{k}\langle X\rangle_{A}$ is essential in
$M\langle X\rangle_{A}$, whence ${\rm udim}(M\langle X\rangle_{A})=k$.

If ${\rm udim}(M_{R})=\infty$, there exist non-zero submodules
$N_{1},N_{2},\ldots$ of $M_{R}$ such that $N_{1}\oplus N_{2}\oplus\cdots$ is a
submodule of $M_{R}$. Thus, every $0\neq N_{i}\langle X\rangle_{A}$ is a
submodule of $M\langle X\rangle_{A}$ for every $i\geq 1$, and $N_{1}\langle
X\rangle_{A}\oplus N_{2}\langle X\rangle_{A}\oplus\cdots$ is a submodule of
$M\langle X\rangle_{A}$, which implies that ${\rm udim}(M\langle
X\rangle_{A})=\infty$. Therefore, ${\rm udim}(M\langle X\rangle_{A})={\rm
udim}(M_{R})$. ∎

######  Corollary 3.17 ([24, Theorem 4.9]).

Let $S:=R[x;\sigma,\delta]$ and $M_{R}$ be a right module. If
$\widehat{M}_{S}$ is good, then ${\rm udim}(\widehat{M}_{S})={\rm
udim}(M_{R})$.

##  4\. Associated primes ideals of induced modules

In this section, we characterize the associated primes of induced modules over
skew PBW extensions (c.f. [31, Section 3]). We recall that a right module
$N_{R}$ is called prime if $N_{R}\neq 0$ and ${\rm ann}_{R}(N_{R})={\rm
ann}_{R}(N_{R}^{{}^{\prime}})$, for every non-zero submodule
$N_{R}^{{}^{\prime}}\subseteq N_{R}$ [5, Definition 1.1]; an ideal $P$ of $R$
is said to be associated of $M_{R}$ if $P$ is prime and there exists a prime
submodule $N_{R}\subseteq M_{R}$ such that $P={\rm ann}_{R}(N_{R})$. The set
of associated prime ideals of $M_{R}$ is denoted by ${\rm Ass}(M_{R})$ [5,
Definition 1.2]. Following Leroy and Matczuk [24], it may happen that ${\rm
Ass}(M_{R})$ is not empty but ${\rm Ass}(N_{R})=\emptyset$, for some non-zero
submodule $N_{R}$ of $M_{R}$ [24, p. 2756]. For this reason, they worked with
modules where ${\rm Ass}(N_{R})$ is not empty for all non-zero submodule
$N_{R}$ of $M_{R}$ and introduced the following definition: $M_{R}$ has enough
prime submodules if any non-zero submodule $N_{R}$ of $M_{R}$ contains a prime
submodule [24, Definition 5.1].

The following lemma shows that if $M_{R}$ has enough prime submodules, then
any non-zero submodule of the induced module $M\langle X\rangle_{A}$ contains
a good polynomial. Lemma 4.1 generalizes [24, Lemma 5.4].

######  Lemma 4.1.

Let $A$ be a bijective skew PBW extension over $R$ and $M_{R}$ be a right
module. If $M_{R}$ has enough prime submodules, then any non-zero submodule
$N_{A}$ of $M\langle X\rangle_{A}$ contains a good polynomial $m$, with
leading coefficient $m_{k}\neq 0$, such that $m_{k}R$ is a prime submodule of
$M_{R}$.

###### Proof.

Let $N_{A}$ be a submodule of $M\langle X\rangle_{A}$ and
$n=n_{1}x^{\alpha_{1}}+\cdots+n_{k}x^{\alpha_{k}}\in N_{A}$ be a non-zero
polynomial with ${\rm lc}(n)=n_{k}\neq 0$ and minimal leading monomial in
$nA$, that is, $x^{\alpha_{k}}\prec{\rm lm}(f)$, for every $f\in nA$. By Lemma
3.2, $n$ is a good polynomial, and since $M_{R}$ contains enough prime
submodules, $n_{k}R$ contains a non-zero prime submodule $m_{k}R$ where
$m_{k}=n_{k}r\neq 0$, for some $r\in R$. Thus, the polynomial
$m=n\sigma^{-\alpha_{k}}(r)\in N_{A}$ is good with leading coefficient $m_{k}$
such that $m_{k}R$ is a prime submodule of $M_{R}$. ∎

Leroy and Matczuk characterized certain right annihilators of generated
modules on $R[x;\sigma,\delta]$ where $\delta$ is a $\sigma$-derivation
$q$-quantized of $R$ [24, Lemma 5.6]. Goodearl and Letzter [17] introduced the
notion of $q$-quantized derivation in the following way: a $\sigma$-derivation
$\delta$ of $R$ is $q$-quantized if $\delta\sigma=q\sigma\delta$ where $q$ is
a central, invertible element of $R$ such that $\sigma(q)=q$ and
$\delta(q)=0$. The ring $R[x;\sigma,\delta]$ is called a $q$-skew polynomial
ring if $\delta$ is $q$-quantized [17, p. 10]. We consider the following
definition for a finite family of endomorphims $\Sigma$ and a family of
$\Sigma$-derivations $\Delta$ of $R$.

######  Definition 4.2.

Let $R$ be a ring and $\Sigma$ be a finite family of endomorphisms of $R$. A
family of $\Sigma$-derivations $\Delta$ of $R$ is called quantized if there
exists $(q_{1},\ldots,q_{n})\in R^{n}$ such that
$\delta_{i}\sigma_{i}=q_{i}\sigma_{i}\delta_{i}$, $\sigma_{i}(q_{j})=q_{j}$
and $\delta_{i}(q_{j})=0$, for every $0\leq i,j\leq n$ where $q_{i}$ is a
central and invertible element of $R$ for every $1\leq i\leq n$.

We say that a skew PBW extension $A$ over a ring $R$ is quantized if the
family of $\Sigma$-derivations $\Delta$ defined in Proposition 2.2 is
quantized.

######  Example 4.3.

We present some examples of quantized skew PBW extensions.

  1. (1)

The algebra of $q$-differential operators $D_{q,h}[x,y]$: Let $q,h\in\Bbbk$,
$q\neq 0$. We consider $\Bbbk[y][x;\sigma,\delta]$ with $\sigma(y):=qy$ and
$\delta(y):=h$. By definition of skew polynomial ring,
$xy=\sigma(y)x+\delta(y)=qyx+h$, and so $xy-qyx=h$. We can prove that
$D_{q,h}[x,y]\cong\sigma(\Bbbk[y])\left\langle x\right\rangle$. It is not
difficult to verify that $\delta\sigma=\sigma\delta$, whence $D_{q,h}[x,y]$ is
a quantized skew PBW extension over $\Bbbk[y]$.

  2. (2)

Additive analogue of the Weyl algebra: Let $\Bbbk$ be a field and
$A_{n}(q_{1},\dots,q_{n})$ be the $\Bbbk$-algebra generated by
$x_{1},\dots,x_{n},t_{1},\dots,t_{n}$ and subject to the relations:

$\displaystyle x_{j}x_{i}$ $\displaystyle=x_{i}x_{j},\ \ \
t_{j}t_{i}=t_{i}t_{j},\ \ \ 1\leq i,j\leq n.$ $\displaystyle x_{j}t_{i}$
$\displaystyle=t_{i}x_{j},\ \ \ i\neq j.$ $\displaystyle x_{i}t_{i}$
$\displaystyle=q_{i}t_{i}x_{i}+1,\ \ \ 1\leq i\leq n.$

where $q_{i}\in\Bbbk\setminus\left\\{0\right\\}$. Thus,
$A_{n}(q_{1},\dots,q_{n})\cong\sigma(\Bbbk[t_{1},\dots,t_{n}])\left\langle
x_{1},\dots,x_{n}\right\rangle$. Notice that $\sigma_{i}(t_{i})=qt_{i}$ and
$\delta_{i}(t_{i})=1$, for all $1\leq i\leq n$. Some simple computations show
that $\delta_{i}\sigma_{i}=\sigma_{i}\delta_{i}$, and so
$A_{n}(q_{1},\dots,q_{n})$ is a quantized skew PBW extension over
$\Bbbk[t_{1},\dots,t_{n}]$.

The following lemma characterizes the annihilators of generated modules by
good polynomials and generalizes [24, Lemma 5.6].

######  Lemma 4.4.

Let $A$ be a bijective skew PBW extension over $R$, $M_{R}$ be a right module,
and $m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}$ be a good polynomial of
$M\langle X\rangle_{A}$ with leading coefficient $m_{k}\neq 0$. If $P:={\rm
ann}_{R}(m_{k}R)$, then:

  * (1)

If $P$ is $(\Sigma,\Sigma^{-1},\Delta)$-stable, then ${\rm
ann}_{A}(mA)=P\langle X\rangle$.

  * (2)

If $A$ is quantized and $P$ is $(\Sigma,\Sigma^{-1})$-stable, then ${\rm
ann}_{A}(mA)=P_{\Delta}\langle X\rangle$

  * (3)

If $A$ is quantized, $m_{k}R$ is prime, and $mA$ contains good polynomial of
any monomial greater than $x^{\alpha_{k}}$, then ${\rm
ann}_{A}(mA)=\sigma^{-\alpha_{k}}(P_{\Sigma,\Delta})\langle X\rangle$.

###### Proof.

  * (1)

Since $P$ is $\Delta$-stable, we have ${\rm ann}_{A}(mA)=P_{\Sigma}\langle
X\rangle$ by Lemma 3.8 (4). Since $(\Sigma,\Sigma^{-1})$-stable, we have
$P_{\Sigma}=P$, and so ${\rm ann}_{A}(mA)=P\langle X\rangle$.

  * (2)

If $r\in P_{\Delta}$, then $\delta^{\beta}(r)\in P$, and so
$\sigma^{\alpha}(\delta^{\beta}(r))\in P$ for every
$\alpha,\beta\in\mathbb{N}^{n}$ by the $(\Sigma,\Sigma^{-1})$-stability of
$P$. If $A$ is quantized,
$\delta^{\beta}(\sigma^{\alpha}(r))=r_{\alpha,\beta}\sigma^{\alpha}(\delta^{\beta}(r))\in
P$ for some $r_{\alpha,\beta}\in R$ whence $P_{\Delta}$ is a
$\Sigma$-invariant ideal of $R$. Hence, $P_{\Delta}\langle X\rangle$ is a two-
sided ideal of $A$ and so $P_{\Delta}\langle X\rangle\subseteq{\rm
ann}_{A}(mA)$. Let $f=b_{1}x^{\beta_{1}}+\cdots+b_{t}x^{\beta_{t}}$ be an
element of ${\rm ann}_{A}(mA)$. By induction on the monomials, we show that
$\delta^{\theta}(b_{i})\in P$, for any $\theta\in\mathbb{N}^{n}$ and $1\leq
i\leq t$. Since $m$ is a good polynomial and $P$ is a
$(\Sigma,\Sigma^{-1})$-stable ideal of $R$, ${\rm
ann}_{A}(mR)=\sigma^{-\alpha_{k}}(P)\langle X\rangle=P\langle X\rangle$ by
Lemma 3.2, and thus $b_{i}\in P$ for every $1\leq i\leq t$. Assume that for
any leading monomial $x^{\gamma}$ with $x^{\gamma}\prec x^{\theta}$, we have
$\delta^{\gamma}(b_{i})\in P$. If $f\in{\rm ann}_{A}(mA)$ and ${\rm
ann}_{A}(mR)=P\langle X\rangle_{A}$, then $mrx^{\theta}f=0$. Since $A$ is
quantized, it follows that
$x^{\theta}b_{i}=r_{1}x^{\theta_{1}}+\cdots+r_{s}x^{\theta_{s}}$ where each
$r_{j}$ is a finite sum of several evaluations of $\sigma^{\theta_{j}}$’s and
$\delta^{\theta-\theta_{j}}$’s in the element $b_{i}$, for every $1\leq j\leq
s$. Thus, if $P$ is a $(\Sigma,\Sigma^{-1})$-stable ideal and
$\delta^{\gamma}(b_{i})\in P$ for any $\gamma\in\mathbb{N}^{n}$ with
$x^{\gamma}\prec x^{\theta}$, then $mRr_{i}x^{\theta_{j}}=0$ for all $1\leq
j\leq s$ where $\theta_{j}\neq 0$. In this way, we have
$mrx^{\theta}b_{i}=mr\delta^{\theta}(b_{i})$, and hence
$mrx^{\theta}f=mr(\delta^{\theta}(b_{1})x^{\beta_{1}}+\cdots+\delta^{\theta}(b_{t})x^{\beta_{t}})$
which shows that $\delta^{\theta}(b_{i})\in P$ for all $1\leq i\leq t$, and
therefore $f\in P_{\Delta}\langle X\rangle$.

  * (3)

Let $f=b_{1}x^{\beta_{1}}+\cdots+b_{t}x^{\beta_{t}}\in{\rm ann}_{A}(mA)$. We
prove by induction on the leading monomials that
$\delta^{\theta}(b_{i})\in\sigma^{-\alpha_{k}}(P_{\Sigma})$, for all
$\theta\in\mathbb{N}^{n}$ and $1\leq i\leq t$. Since $mA$ contains good
polynomials of any monomial greater than $x^{\alpha_{k}}$, there exists
$f_{\gamma_{i}}\in mA$ with leading monomial $x^{\gamma_{i}}$, for each
$x^{\gamma_{i}}\succeq x^{\alpha_{k}}$. Let $m_{\gamma_{i}}$ be the leading
coefficient of $f_{\gamma_{i}}$ where $m_{\gamma_{i}}\in m_{k}R$. If $m_{k}R$
is prime, then $m_{\gamma_{i}}R$ is prime, and thus ${\rm
ann}(m_{\gamma_{i}}R)=P$ for all $\gamma_{i}$. Since the $f_{\gamma_{i}}$ are
good polynomials, ${\rm
ann}_{A}(f_{\gamma_{i}}R)=\sigma^{-\gamma_{i}}(P)\langle X\rangle$ by Lemma
3.2. If $E$ denotes the submodule of $M\langle X\rangle_{R}$ defined by
$E=\sum_{\gamma_{i}\geq\alpha_{k}}f_{\gamma_{i}}R$, then $E\subseteq mA$ and
${\rm ann}_{A}(mA)\subseteq{\rm
ann}_{A}(E)=\bigcap_{\gamma_{i}\geq\alpha_{k}}\sigma^{-\gamma_{i}}(P)\langle
X\rangle=\sigma^{-\alpha_{k}}(P_{\Sigma})\langle X\rangle$. In this way, if
$f\in{\rm ann}_{A}(mA)$, then $f\in\sigma^{-\alpha_{k}}(P_{\Sigma})\langle
X\rangle$ and thus $b_{i}\in\sigma^{-\alpha_{k}}(P_{\Sigma})$.

Now, assume that for any leading monomial $x^{\gamma}$ with $x^{\gamma}\prec
x^{\theta}$, we have
$\delta^{\gamma}(b_{i})\in\sigma^{-\alpha_{k}}(P_{\Sigma})$. If
$Ex^{\theta}\subseteq mA$ and
$f=b_{1}x^{\beta_{1}}+\cdots+b_{t}x^{\beta_{t}}\in{\rm ann}_{A}(mA)$, then
$Ex^{\theta}f=0$. In addition if $A$ is quantized, then
$x^{\theta}b_{i}=r_{1}x^{\theta_{1}}+\cdots+r_{s}x^{\theta_{s}}$ where each
$r_{j}$ is a finite sum of several evaluations of $\sigma^{\theta_{j}}$’s and
$\delta^{\theta-\theta_{j}}$’s in the element $b_{i}$, for every $1\leq j\leq
s$. If $\delta^{\gamma}(b_{i})\in\sigma^{-\alpha_{k}}(P_{\Sigma})$ for any
$\gamma\in\mathbb{N}^{n}$ with $x^{\gamma}\prec x^{\theta}$, then
$Er_{i}x^{\theta_{j}}=0$ for all $1\leq j\leq s$ where $\theta_{j}\neq 0$.
Thus $Ex^{\theta}b_{i}=E\delta^{\theta}(b_{i})$, and hence
$Ex^{\theta}f=E(\delta^{\theta}(b_{1})x^{\beta_{1}}+\cdots+\delta^{\theta}(b_{t})x^{\beta_{t}})$
whence $\delta^{\theta}(b_{i})\in\sigma^{-\alpha_{k}}(P_{\Sigma})$ for all
$1\leq i\leq t$. So ${\rm
ann}_{A}(mA)\subseteq(\sigma^{-\alpha_{k}}(P_{\Sigma}))_{\Delta}A$, and since
$A$ is quantized, it follows that
$(\sigma^{-\alpha_{k}}(P_{\Sigma}))_{\Delta}=\sigma^{-\alpha_{k}}(P_{\Sigma,\Delta})$,
and hence ${\rm ann}_{A}(mA)\subseteq\sigma^{-\alpha_{k}}(P_{\Sigma,\Delta})$.

To prove the other inclusion, if
$r\in\sigma^{-\alpha_{k}}(P_{\Sigma,\Delta})$, then $\sigma^{\alpha_{k}}(r)\in
P_{\Sigma,\Delta}$, and thus
$\sigma^{\alpha_{k}}(\sigma^{\alpha}\delta^{\beta}(r))$ for all
$\alpha,\beta\in\mathbb{N}$. Since $m$ is a good polynomial of leading
monomial $x^{\alpha_{k}}$ and leading coefficient $m_{k}\neq 0$, then
$mR\sigma^{\alpha}\delta^{\beta}(r)=0$. This implies that $mRx^{\gamma}r=0$,
for any $\gamma\in\mathbb{N}^{n}$, and so
$\sigma^{-\alpha_{k}}(P_{\Sigma,\Delta})\subseteq{\rm ann}_{A}(mA)$.

∎

The following theorem characterizes the associated prime ideals of $M\langle
X\rangle_{A}$ where $M_{R}$ is a right module that contains enough prime
submodules.

######  Theorem 4.5.

Let $A$ be a bijective skew PBW extension over $R$, $M_{R}$ be a right module,
and $Q\in{\rm Ass}(M\langle X\rangle_{A})$. If $M_{R}$ contains enough prime
submodule, then:

  * (1)

If $P$ is $(\Sigma,\Sigma^{-1},\Delta)$-stable for every $P\in{\rm
Ass}(M_{R})$, then $Q=P\langle X\rangle$ for some $P\in{\rm Ass}(M_{R})$.

  * (2)

If $A$ is quantized and $P$ is $(\Sigma,\Sigma^{-1})$-stable for every
$P\in{\rm Ass}(M_{R})$, then $Q=P_{\Delta}\langle X\rangle$ for some $P\in{\rm
Ass}(M_{R})$.

  * (3)

If $A$ is quantized and the module $M\left\langle X\right\rangle_{A}$ is good,
then $Q=P_{\Sigma,\Delta}\langle X\rangle$ for some $P\in{\rm Ass}(M_{R})$ and
$P_{\Sigma,\Delta}$ is $(\Sigma,\Sigma^{-1})$-stable.

###### Proof.

  * (1)

Let $N_{A}$ be a prime submodule of $M\langle X\rangle_{A}$ such that ${\rm
ann}_{A}(N)=Q$. By Lemma 4.1, there is a good polynomial
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in N_{A}$ with leading
coefficient $m_{k}\neq 0$ such that $m_{k}R$ is prime submodule of $M_{R}$.
Since $N_{A}$ is prime, then $mA$ is also prime and ${\rm ann}_{A}(mA)={\rm
ann}_{A}(N)=Q$. By Lemma 4.4 (1), we have $Q=P\langle X\rangle$ with $P={\rm
ann}_{R}(m_{k}R)\in{\rm Ass}(M_{R})$.

  * (2)

In the same way, if $N_{A}$ is a prime submodule of $M\langle X\rangle_{A}$
with ${\rm ann}_{A}(N)=Q$, there exists a good polynomial
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in N_{A}$ with leading
coefficient $m_{k}\neq 0$ such that $m_{k}R$ is prime submodule of $M_{R}$ by
Lemma 4.1. Since $N_{A}$ is prime, then $mA$ is also prime and ${\rm
ann}_{A}(mA)={\rm ann}_{A}(N)=Q$. By Lemma 4.4 (2), $Q=P_{\Delta}\langle
X\rangle$ where $P={\rm ann}_{R}(m_{k}R)\in{\rm Ass}(M_{R})$.

  * (3)

If $N_{A}$ is a prime submodule of $M\langle X\rangle_{A}$ such that ${\rm
ann}_{A}(N)=Q$, there exists a good polynomial
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}\in N_{A}$ with leading
coefficient $m_{k}\neq 0$ by Lemma 4.1. If $M\langle X\rangle_{A}$ is good,
$mA$ contains a good polynomial $m^{\prime}$ with leading monomial $x^{\beta}$
such that $x^{\alpha_{k}}\preceq x^{\beta}$. Additionally, $mA$ is prime
module which implies that ${\rm ann}_{A}(mA)=Q$, and by Lemma 4.4 (3), ${\rm
ann}_{A}(m^{\prime}A)=\sigma^{-\beta}(P_{\Sigma,\Delta})\langle X\rangle$ and
${\rm ann}_{A}(mA)=\sigma^{-\alpha_{k}}(P_{\Sigma,\Delta})\langle X\rangle$.
Thus, $P_{\Sigma,\Delta}$ is $(\Sigma,\Sigma^{-1})$-stable, and hence $Q={\rm
ann}_{A}(mA)=P_{\Sigma,\Delta}\langle X\rangle$.

∎

######  Corollary 4.6 ([24, Theorem 5.7]).

Let $S:=R[x;\sigma,\delta]$ and $M_{R}$ be a right module. If $M_{R}$ contains
enough prime submodule and $Q\in{\rm Ass}(\widehat{M}_{S})$, then:

  * (1)

If for every $P\in{\rm Ass}(M_{R})$, $\sigma(P)=P$ and $\delta(P)\subseteq P$,
then $Q=PS$ for some $P\in{\rm Ass}(M_{R})$.

  * (2)

If $\delta$ is $q$-quantized and $\sigma(P)=P$ for all $P\in{\rm Ass}(M_{R})$,
then $Q=P_{\delta}S$ for some $P\in{\rm Ass}(M_{R})$.

  * (3)

If $\delta$ is $q$-quantized and $\widehat{M}_{S}$ is a good module, then
$Q=P_{\sigma,\delta}S$ for some $P\in{\rm Ass}(M_{R})$ and
$\sigma(P_{\sigma,\delta})=P_{\sigma,\delta}$.

Leroy and Matczuk presented an example where Corollary 4.6 fails if the module
$M_{R}$ does not have enough prime submodules. This shows that this hypothesis
is not superfluous. If $M$ is a $\Bbbk$-linear space with basis
$\\{v_{i}\\}_{i\in\mathbb{Z}}$ and $R=\Bbbk\langle X\rangle$ is the free
algebra over $\Bbbk$ on the set $X=\\{x_{i}\\}_{i\in\mathbb{Z}}$, then $M$ has
a module structure over $R$ given by $v_{i}x_{k}=v_{i+1}$ if $i\leq k$ and $0$
otherwise. Let $\sigma$ be the automorphism of $R$ defined by
$\sigma(x_{k})=x_{k+1}$ for any $k\in\mathbb{Z}$ and $S:=R[t,\sigma]$. They
showed that ${\rm Ass}(M_{R})=\emptyset$ and that $\widehat{M}_{S}$ is prime
with ${\rm Ass}(\widehat{M}_{S})=\\{0\\}$ [24, Example 5.8].

The following theorem shows when $M\langle X\rangle_{A}$ is a prime module and
characterizes its associated prime ideals in terms of the associated primes of
$M_{R}$.

######  Theorem 4.7.

Let $A$ be a bijective skew PBW extension over $R$ and $M_{R}$ be a right
module. If $M_{R}$ is a prime module with $P={\rm ann}_{R}(M)$, then:

  * (1)

If $P$ is $(\Sigma,\Sigma^{-1},\Delta)$-stable, then the induced module
$M\langle X\rangle_{A}$ is prime with the associated prime ideal equal to
$Q=P\langle X\rangle$.

  * (2)

If $A$ is quantized and $P$ is $(\Sigma,\Sigma^{-1})$-stable, then $M\langle
X\rangle_{A}$ is prime with the associated prime ideal equal to
$Q=P_{\Delta}\langle X\rangle$.

  * (3)

If $A$ is quantized and the module $M\langle X\rangle_{A}$ is good, then
$M\langle X\rangle_{A}$ is a prime module if and only if $P_{\Sigma,\Delta}$
is $(\Sigma,\Sigma^{-1})$-stable. If $M\langle X\rangle_{A}$ is prime, then
its associated prime ideal is equal to $Q=P_{\Sigma,\Delta}\langle X\rangle$.

###### Proof.

  * (1)

Let $N_{A}$ be a submodule of $M\langle X\rangle_{A}$ and
$m=m_{1}x^{\alpha_{1}}+\cdots+m_{k}x^{\alpha_{k}}$ be an element of $N_{A}$
with minimal leading monomial in $mA_{A}$, i.e., $x^{\alpha_{k}}\preceq{\rm
lm}(f)$ for all $f\in mA_{A}$. By Lemma 3.2, $m$ is a good polynomial and
since $M_{R}$ is prime, then $m_{k}R$ is a submodule prime of $M_{R}$ with
$P={\rm ann}_{R}(m_{k}R)$ by Lemma 4.1. Additionally, by Lemma 4.4 (1), we
have $P\langle X\rangle={\rm ann}_{A}(mA)$, and thus $P\langle
X\rangle\subseteq{\rm ann}_{A}(M\langle X\rangle)\subseteq{\rm
ann}_{A}(N)\subseteq{\rm ann}_{A}(mA)=P\langle X\rangle$. This implies that
${\rm ann}_{A}(N)=P\langle X\rangle$ for any submodule $N_{A}$ of $M\langle
X\rangle_{A}$, whence $M\langle X\rangle_{A}$ is a prime module with
associated prime ideal $Q=P\langle X\rangle$.

  * (2)

Following the same argument in (1) and Lemma 4.4 (2), $M\langle X\rangle_{A}$
is a prime module with associated prime ideal $Q=P_{\Delta}\langle X\rangle$.

  * (3)

In the same way, the argument of (1) and Lemma 4.4 (3) show that $M\langle
X\rangle_{A}$ is a prime module with associated prime ideal
$Q=P_{\Sigma,\Delta}\langle X\rangle$ and $P_{\Sigma,\Delta}$ is
$(\Sigma,\Sigma^{-1})$-stable.

∎

######  Corollary 4.8 ([24, Theorem 5.10]).

Let $S:=R[x;\sigma,\delta]$ and $M_{R}$ be a right module. If $M_{R}$ is prime
with $P={\rm ann}_{R}(M)$, then:

  * (1)

Suppose that $\sigma(P)=P$ and $\delta(P)\subseteq P$. Then the induced module
$\widehat{M}_{S}$ is prime with the associated prime ideal equal to $PS=Q$.

  * (2)

Suppose that $\delta$ is a $q$-quantized $\sigma$-derivation and
$\sigma(P)=P$. Then $\widehat{M}_{S}$ is prime with the associated prime ideal
equal to $P_{\delta}S=Q$.

  * (3)

Suppose that $\delta$ is a $q$-quantized $\sigma$-derivation and the module
$\widehat{M}_{S}$ is good. Then $\widehat{M}_{S}$ is a prime module if and
only if $\sigma(P_{\sigma,\delta})=P_{\sigma,\delta}$. Moreover, if
$\widehat{M}_{S}$ is prime, then its associated prime ideal is equal to
$P_{\sigma,\delta}S=Q$.

##  5\. Examples

The importance of our results is appreciated when we apply them to algebraic
structures more general than those considered by Leroy and Matczuk [24], that
is, some noncommutative rings which cannot be expressed as skew polynomial
rings. In this section, we consider several families of rings that have been
studied in the literature which are subfamilies of skew PBW extensions.

######  Example 5.1.

[14, p. 30] The diffusion algebra $A$ is generated by $2n$ indeterminates
$D_{i},x_{i}$ over $\Bbbk$ with $1\leq i\leq n$ and subjects to the relations

$\displaystyle x_{i}x_{j}=x_{j}x_{i},\ \ x_{i}D_{j}=D_{j}x_{i},\ \ 1\leq
i,j\leq n$,

$c_{ij}D_{i}D_{j}-c_{ji}D_{j}D_{i}=x_{j}D_{i}-x_{i}D_{j},\ \ i<j,\ \
c_{ij},c_{ji}\in\Bbbk^{*}$.

According to Definition 2.1, the algebra $A$ can be seen as a skew PBW
extension over $\Bbbk[x_{1},\dots,x_{n}]$, but not as a PBW extension or an
iterated skew polynomial ring of injective type. If $M_{R}$ is a right module
over $R:=\Bbbk[x_{1},\dots,x_{n}]$ and $M\langle
D_{1},\ldots,D_{n}\rangle_{A}$ is a good module, then Theorem 3.16 shows that
$M_{R}$ and $M\langle D_{1},\ldots,D_{n}\rangle_{A}$ have the same uniform
dimension. If $M_{R}$ contains enough prime submodules, then Theorem 4.5
characterizes the associated prime ideals of $M\langle
D_{1},\ldots,D_{n}\rangle_{A}$, and if $M_{R}$ is a prime module with $P={\rm
ann}_{R}(M)$, then $M\langle D_{1},\ldots,D_{n}\rangle_{A}$ is prime module
with associated prime ideal $P\langle D_{1},\ldots,D_{n}\rangle$ by Theorem
4.7.

######  Example 5.2.

[12, Section 25.2] The basis of the Lie 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
$[\bf{J}_{\alpha\beta},\bf{J}_{\mu\nu}]=\delta_{\beta\mu}\bf{J}_{\alpha\nu}+\delta_{\alpha\nu}\bf{J}_{\beta\mu}-\delta_{\beta\nu}\bf{J}_{\alpha\mu}-\delta_{\alpha\mu}\bf{J}_{\beta\nu}$.
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 $\mathbb{C}$ [9,
Section 5], then $U(\mathfrak{so}(5,\mathbb{C}))$ is a skew PBW extension over
$\mathbb{C}$, i.e.,
$U(\mathfrak{so}(5,\mathbb{C}))\cong\sigma(\mathbb{C})\langle{\bf
J}_{\alpha\beta}\mid 1\leq\alpha\leq\beta\leq 5\rangle$. If $M_{\mathbb{C}}$
is a right module over $\mathbb{C}$ and $M\langle{\bf
J}_{\alpha\beta}\rangle_{A}$ is a good module, then Theorem 3.16 characterizes
the uniform dimension of the module $M\langle{\bf J}_{\alpha\beta}\rangle_{A}$
over $A:=U(\mathfrak{so}(5,\mathbb{C}))$, and if $M_{\mathbb{C}}$ contains
enough prime submodules, then Theorem 4.5 describes the associated prime
ideals of $M\langle{\bf J}_{\alpha\beta}\rangle_{A}$.

######  Example 5.3.

Following Havliček et al. [18, p. 79], the $\mathbb{C}$-algebra
$U_{q}^{\prime}(\mathfrak{so}_{3})$ is generated 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 $0\neq q\in\mathbb{C}$. It is straightforward to show that
$U_{q}^{\prime}(\mathfrak{so}_{3})$ cannot be expressed as an iterated skew
polynomial ring. However, this algebra can be seen as a skew PBW extension
over $\mathbb{C}$ [14, Example 1.3.3]. If $M_{\mathbb{C}}$ is a right module
over $\mathbb{C}$ and $M\langle I_{1},I_{2},I_{3}\rangle_{A}$ is a good
module, then Theorem 3.16 characterizes the uniform dimension of $M\langle
I_{1},I_{2},I_{3}\rangle_{A}$ with $A:=U_{q}^{\prime}(\mathfrak{so}_{3})$. If
$M_{\mathbb{C}}$ contains enough prime submodules, then Theorem 4.5 describes
the associated prime ideals of $M\langle I_{1},I_{2},I_{3}\rangle_{A}$, and if
$M_{\mathbb{C}}$ is a prime module with $P={\rm ann}_{\mathbb{C}}(M)$, then
$M\langle I_{1},I_{2},I_{3}\rangle_{A}$ is a prime module with associated
prime ideal $P\langle I_{1},I_{2},I_{3}\rangle$ by Theorem 4.7.

######  Example 5.4.

Zhedanov [44] defined the Askey-Wilson algebra AW(3) as the algebra generated
by three indeterminates $K_{0},K_{1}$, and $K_{2}$, subject to the commutation
relations given by

$\displaystyle e^{\omega}K_{0}K_{1}-e^{-\omega}K_{1}K_{0}$
$\displaystyle=K_{2},e^{\omega}K_{2}K_{0}-e^{-\omega}K_{0}K_{2}=BK_{0}+C_{1}K_{1}+D_{1},$
$\displaystyle e^{\omega}K_{1}K_{2}-e^{-\omega}K_{2}K_{1}$
$\displaystyle=BK_{1}+C_{0}K_{0}+D_{0},$

where $B,C_{0},C_{1},D_{0},D_{1}\in\mathbb{R}$ and $\omega$ is an arbitrary
real parameter. It is not difficult to see that AW(3) cannot be expressed as
an iterated skew polynomial ring. On the other hand, using techniques such as
those presented in [14, Theorem 1.3.1], it can be shown that ${\rm AW(3)}$ is
a skew PBW extension of endomorphism type over $\mathbb{R}$, that is, ${\rm
AW(3)}\cong\sigma(\mathbb{R})\langle K_{0},K_{1},K_{2}\rangle$. If
$M_{\mathbb{R}}$ is a right module over $\mathbb{R}$ and $M\langle
K_{0},K_{1},K_{2}\rangle_{A}$ is a good module over $A:=AW(3)$, then ${\rm
udim}(M\langle K_{0},K_{1},K_{2}\rangle_{A})={\rm udim}(M_{\mathbb{R}})$ by
Theorem 3.16. If $M_{\mathbb{R}}$ contains enough prime submodules, then
Theorem 4.5 characterizes the associated prime ideals of $M\langle
K_{0},K_{1},K_{2}\rangle_{A}$, and if $M_{\mathbb{R}}$ is a prime module with
$P={\rm ann}_{\mathbb{R}}(M)$, then $M\langle K_{0},K_{1},K_{2}\rangle_{A}$ is
a prime module with associated prime ideal $P\langle K_{0},K_{1},K_{2}\rangle$
by Theorem 4.7.

##  6\. Future work

As a possible future work, we have in mind to study the couniform dimension of
modules introduced by Varadarajan [39, 43]. Annin [6] studied this dimension
on the inverse polynomial module $M[x^{-1}]$ over $R[x;\sigma]$. We think that
a natural task is to investigate the counifom dimension of the polynomial
module $M[x^{-1}]$ on structures more general than skew polynomial rings of
automorphism type.

Macdonald [28] introduced the dual notion of primary decomposition known as
secondary representation, where the main ideals of his theory are called
attached primes. Since Annin [7] characterized the attached prime ideals on
the inverse polynomial module $M[x^{-1}]$ over $R[x;\sigma]$, another natural
task is to investigate the attached prime ideals of modules $M[x^{-1}]$ over
structures more general than skew polynomial rings of automorphism type (for
instance, the skew PBW extensions and the semi-graded rings introduced by
Lezama and Latorre [23]).

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