# On $\Sigma$-skew reflexive-nilpotents-property for rings

Héctor Suárez  Universidad Pedagógica y Tecnológica de Colombia, Sede Tunja
Campus Universitario<EMAIL_ADDRESS>,  Sebastián Higuera
Universidad Nacional de Colombia - Sede Bogotá Campus Universitario
<EMAIL_ADDRESS>and  Armando Reyes  Universidad Nacional de Colombia -
Sede Bogotá Campus Universitario<EMAIL_ADDRESS>Dedicated to the memory
of Professor Vyacheslav Artamonov

###### Abstract.

In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for
Ore extensions of injective type, and more generally, skew PBW extensions.
With this aim, we introduce the notions of $\Sigma$-skew CN rings and
$\Sigma$-skew reflexive (RNP) rings, for $\Sigma$ a finite family of ring
endomorphisms of a ring $R$. Under certain conditions of compatibility, we
study the transfer of the $\Sigma$-skew RNP property from a ring of
coefficients to an Ore extension or skew PBW extension over this ring. We also
consider this property for localizations of these noncommutative rings. Our
results extend those corresponding presented by Bhattacharjee [9].

###### Key words and phrases:

RNP ring, skew polynomial ring, skew PBW extension

###### 2020 Mathematics Subject Classification:

16S36, 16S80, 16U99, 16W20, 16N40

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

##  1\. Introduction

Throughout the paper, $\mathbb{N}$ and $\mathbb{Z}$ denote the set of natural
numbers including zero and the ring of integers, respectively. The symbol
$\Bbbk$ denotes a field, and $\Bbbk^{*}:=\Bbbk\ \backslash\ \\{0\\}$. Every
ring is associative with identity unless otherwise stated. For a ring $R$,
$N(R)$ denotes its set of nilpotent elements, $N_{*}(R)$ is its prime radical,
and $N^{*}(R)$ is its upper nilradical (i.e. the sum of all nil ideals). It is
well-known that $N^{*}(R)\subseteq N(R)$, and if the equality $N^{*}(R)=N(R)$
holds, then $R$ is called NI (Marks [52]). Note that by definition, $R$ is NI
if and only if $N(R)$ forms an ideal if and only if $R/N^{*}(R)$ is reduced,
that is, $R/N^{*}(R)$ has no nonzero nilpotent elements. Hong and Kwak [30,
Corollary 13], showed that a ring $R$ is NI if and only if every minimal
strongly prime ideal of $R$ is completely prime. If the equality
$N_{*}(R)=N(R)$ holds, then $R$ is called 2-primal (Birkenmeier et al. [10]);
equivalently, $N^{*}(R)$ is a completely semiprime ideal of $R$. 2-primal
rings are clearly NI. The converse of this implication need not hold, but if
$R$ is an NI ring of bounded index nilpotency, then $R$ is 2-primal [32,
Proposition 1.4].

Following Cohn [14], $R$ is said to be reversible if $ab=0$ implies $ba=0$,
for $a,b\in R$. Commutative rings and reduced rings are clearly reversible.
Reversible rings were studied under the name zero commutative by Habeb [20].
As a matter of fact, the class of NI rings contains nil rings and reversible
rings. Lambek [43] called a ring $R$ symmetric provided $abc=0$ implies
$acb=0$, for $a,b,c\in R$. Of course, commutative rings are symmetric, and
symmetric rings are reversible, but the converses do not hold ([2, Examples
I.5 and II.5], and [53, Examples 5 and 7]). We know that every reduced ring is
symmetric [66, Lemma 1.1], but the converse does not hold [2, Example II.5].
Bell [7] used the term _Insertion-of-Factors-Property_ (IFP) for a ring $R$ if
$ab=0$ implies $aRb=0$, for $a,b\in R$ (Narbonne [56] and Habeb [20] used the
terms semicommutative and zero insertive, respectively). Some results about
IFP rings are due to Shin [66]. Note that every reversible ring is IFP, but
the converse need not hold [38, Lemma 1.4 and Example 1.5]. Reversible rings
are reflexive, and there exists a reflexive and IFP ring which is not
symmetric [53, Examples 5 and 7]. In fact, a ring $R$ is reflexive and IFP if
and only if $R$ is reversible [41, Proposition 2.2]. It is easy to see that
IFP rings are 2-primal. Note that IFP rings are Abelian (every idempotent is
central).

In addition to reversible rings and their aforementioned generalizations of
reduced rings, we also have the notion of reflexive ring. Mason [54]
introduced the reflexive property for right ideals by defining a right ideal
$I$ of a ring $R$ as reflexive if for $a,b\in R$, $aRb\subseteq I$ implies
$bRa\subseteq I$, and a ring $R$ is called reflexive if the zero ideal of $R$
is reflexive (i.e. $aRb=0$ implies $bRa=0$, for $a,b\in R$). Equivalently, $R$
is reflexive if and only if $IJ=0$ implies $JI=0$, for ideals $I,J$ of $R$
[41, Lemma 2.1]. By a direct computation one can check that semiprime rings
and reversible rings are reflexive, and every ideal of a fully idempotent ring
$R$ (i.e. $I^{2}=I$, for every ideal of $R$) is reflexive (Courter [15]). By
[41, Example 2.3], we know that the IFP and the reflexive ring properties do
not imply each other. Kwak and Lee [41] characterized the aspects of the
reflexive and one-sided idempotent reflexive properties. They established a
method by which a reflexive ring, which is not semiprime, can always be
constructed from any semiprime ring, and showed that the reflexive property is
Morita invariant. In the literature, different generalizations of reflexive
rings have been formulated. Let us recall them.

Kim [35] introduced the notion of idempotent reflexive ring as a
generalization of reflexive ring. For a one-sided ideal $I$ of a ring $R$, $I$
is called right idempotent reflexive if $aRe\subseteq I$ implies $eRa\subseteq
I$, for any $a,e^{2}=e\in R$, and the ring $R$ is called right idempotent
reflexive if the zero ideal is a right idempotent reflexive ideal. Left
idempotent reflexive ideals and left idempotent reflexive rings are defined
similarly. If a ring $R$ is both left and right idempotent reflexive, then $R$
is called an idempotent reflexive ring. For more details about these rings,
see Kim and Baik [36]. As one can check, reflexive rings and Abelian rings are
idempotent reflexive. By [41, Example 2.3(1)], we know that there exists an
idempotent reflexive ring which is not reflexive. Similar to the case of
reflexive rings, $R$ is right idempotent reflexive if and only if $IJ=0$
implies $JI=0$, for all right ideals $I,J$ of $R$ where $J$ is a right ideal
generated by a subset of idempotents in $R$, if and only if $IJ=0$ implies
$JI=0$, for all ideals $I,J$ of $R$ where $J$ is an ideal generated by a
subset of idempotents in $R$ [41, Lemma 3.4].

Kheradmand et al. [33] also introduced a generalization of reflexive rings. A
ring $R$ is said to be RNP (reflexive-nilpotents-property) if for $a,b\in
N(R)$, $aRb=0$ implies $bRa=0$. Of course, reflexive rings are RNP but the
converse need not hold [34, Example 1.2(1)]. In the same year, Kheradmand et
al. [34] defined a more general class than reflexive and RNP rings by
considering nil ideals, and called a ring $R$ nil-reflexive if $IJ=0$ implies
$JI=0$, for nil ideals $I,J$ of $R$. As it is clear, reflexive rings are RNP
and RNP rings are nil-reflexive, but the converse need not hold [34, Example
1.2]. They also showed that $R$ is a nil-reflexive ring if and only if $aRb=0$
implies $bRa=0$, for elements $a,b\in N^{*}(R)$ [34, Proposition 2.1]. Notice
that the concepts of (nil)reflexive rings and NI rings are independent of each
other [34, Examples 1.5 and 2.9]. Nevertheless, if $R$ is an NI ring, then the
following assertions are equivalent: $R$ is nil-reflexive; $aRb=0$, for
$a,b\in N(R)$, implies $bRa=0$; $IJ=0$ implies $JI=0$, for all nil right (or,
left) ideals $I,J$ of $R$ [34, Proposition 1.6].

In addition to the reduced rings and its generalizations described above, they
have all been extended by ring endomorphisms. According to Krempa [39], an
endomorphism $\sigma$ of a ring $R$ is called rigid if $a\sigma(a)=0$ implies
$a=0$, for $a\in R$, and by Hong et al. [28], $R$ is called $\sigma$-rigid if
there exists a rigid endomorphism $\sigma$ of $R$. Any rigid endomorphism of a
ring $R$ is a monomorphism and $\sigma$-rigid rings are reduced rings [28,
Proposition 5]. Following Başer et al. [6] an endomorphism $\sigma$ of a ring
$R$ is called right skew reversible if whenever $ab=0$, for $a,b\in R$,
$b\sigma(a)=0$, and the ring $R$ is called right $\sigma$-skew reversible if
there exists a right skew reversible endomorphism $\sigma$ of $R$. Similarly,
left $\sigma$-skew reversible rings are defined. A ring $R$ is said to be
$\sigma$-skew reversible if it is both right and left $\sigma$-reversible. It
is important to say that $R$ is a $\sigma$-rigid ring if and only if $R$ is
semiprime and right $\sigma$-skew reversible for a monomorphism $\sigma$ of
$R$ [6, Proposition 2.5 (iii)].

Kwak et al. [42] extended the reflexive property to the skewed reflexive
property by ring endomorphisms. An endomorphism $\sigma$ of a ring $R$ is
called right (resp., left) skew reflexive if for $a,b\in R$, $aRb=0$ implies
$bR\sigma(a)=0$ (resp., $\sigma(b)Ra=0$), and $R$ is called right (resp.,
left) $\sigma$-skew reflexive if there exists a right (resp., left) skew
reflexive endomorphism $\sigma$ of $R$. $R$ is said to be $\sigma$-skew
reflexive if it is both right and left $\sigma$-skew reflexive. It is clear
that $\sigma$-rigid rings are right $\sigma$-skew reflexive. More precisely, a
ring $R$ is reduced and right $\sigma$-skew reflexive for a monomorphism
$\sigma$ of $R$ if and only if $R$ is $\sigma$-rigid [42, Theorem 2.6]. Now,
Bhattacharjee [9] extend the notion of RNP rings to ring endomorphisms
$\sigma$ and introduced the notion of $\sigma$-skew RNP rings as a
generalization of $\sigma$-skew reflexive rings. An endomorphism $\sigma$ of a
ring $R$ is called right (resp., left) skew RNP if for $a,b\in N(R)$, $aRb=0$
implies $bR\sigma(a)=0$ (resp., $\sigma(b)Ra=0$). A ring $R$ is called right
(resp., left) $\sigma$-skew RNP if there exists a right (resp., left) skew RNP
endomorphism $\sigma$ of $R$. $R$ is said to be $\sigma$-skew RNP if it is
both right and left $\sigma$-skew RNP. From [9, Remark 1.2], we know that
reduced rings are $\sigma$-skew RNP for any endomorphism $\sigma$, and every
right (resp., left) $\sigma$-skew reflexive ring is right (resp., left)
$\sigma$-skew RNP. By [9, Example 1.3], we have that the notion of
$\sigma$-skew RNP ring is not left-right symmetric. However, if $R$ is an RNP
ring with an endomorphism $\sigma$, then $R$ is right $\sigma$-skew RNP if and
only if $R$ is left $\sigma$-skew RNP.

Taking into account our interest in noncommutative rings defined by
endomorphisms, in this paper we will focus our attention on the study of ring-
theoretical notions above for the skew polynomial rings (also known as Ore
extensions) defined by Ore [59], and the skew PBW extensions introduced by
Gallego and Lezama [17]. As is well-known, skew polynomial rings are one of
the most important families of noncommutative rings of polynomial type related
with the study of quantum groups, differential operators, noncommutative
algebraic geometry and noncommutative differential geometry (e.g. Brown and
Goodearl [12], Goodearl and Warfield [19] or McConnell and Robson [55]), and a
lot of papers have been published with the aim of studying different
theoretical properties of these objects. Now, regarding skew PBW extensions,
their importance is that these objects generalize PBW extensions defined by
Bell and Goodearl [8], families of differential operator rings, Ore extensions
of injective type, several algebras appearing in noncommutative algebraic
geometry, examples of quantum groups, and other families of noncommutative
rings having PBW bases. Since its introduction, ring-theoretical and
homological properties of skew PBW extensions have been studied by some people
(e.g. Artamonov [5], Hamidizadeh et al. [21], Hashemi et al. [22, 23, 24],
Lezama et al. [45, 46, 49, 50], Tumwesigye et al. [69], and the authors [26,
63], and [67]). As a matter of fact, a book that includes several of the works
carried out for these extensions has been published (Fajardo et al. [16]).

The paper is organized as follows. Section 2 introduces notation, and gives a
brief account of the main results on skew PBW extensions appearing in [16, 17]
which are required in the paper. In Section 3, the notion of $\Sigma$-skew
reflexive is introduced for a family $\Sigma$ of endomorphisms of a ring $R$
in order to generalize the notions of skew reflexive endomorphism and
$\sigma$-skew reflexive ring, and to study different properties for this new
family of rings. In Section 4, we introduce the concept of skew RNP for a
finite family of endomorphisms $\Sigma$ of a ring $R$, and the $\Sigma$-skew
RNP rings as a generalization of $\sigma$-skew reflexive rings and
$\sigma$-skew RNP rings, presented [42] and [9], respectively. We study
properties relating the $\Sigma$-skew RNP condition with different kinds of
rings widely studied in the literature. Also, we present some results that
generalize in some way the results presented in [9] for skew polynomial rings.
Finally, we study the behavior of the $\Sigma$-skew RNP property for Ore
localization by regular elements. In particular, we present a theorem that
characterizes this property for the localization of skew PBW extensions.

##  2\. Preliminaries

######  Definition 2.1.

([17, Definition 1]) Let $R$ and $A$ be rings. We say that $A$ is 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. (1)

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

  2. (2)

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. (3)

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. (4)

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

From the definition it follows that every non-zero element $f\in A$ can be
uniquely expressed as $f=a_{0}+a_{1}X_{1}+\cdots+a_{m}X_{m}$, with $a_{i}\in
R$ and $X_{i}\in\text{Mon}(A)$, for $0\leq i\leq m$ [17, Remark 2]. For
$X=x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\in\text{Mon}(A)$,
$\text{deg}(X)=|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$.

The next result establishes the importance of the (injective) endomorphisms of
the ring of coefficients. As expected, later on we will make these
endomorphisms interact with the different ring-theoretical notions of rings
discussed in Section 1.

######  Proposition 2.2.

([17, Proposition 3]) If $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a
skew PBW extension, 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 $\sigma_{i}$-derivations of Proposition 2.2, respectively.
For a skew PBW extension $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ over
$R$, we say that the pair $(\Sigma,\Delta)$ 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 the classical composition of functions.

######  Definition 2.3.

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

  * (1)

[17, Definition 4] $A$ is called quasi-commutative if the conditions
${\rm(3)}$ and ${\rm(4)}$ presented above are replaced by the following:

    1. (3’)

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. (4’)

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

  * (2)

[17, Definition 4] $A$ is said to be bijective if $\sigma_{i}$ is bijective
for each $1\leq i\leq n$, and $d_{i,j}$ is invertible for any $1\leq i<j\leq
n$.

  * (3)

[47, Definition 2.3] If $\sigma_{i}\in\Sigma$ is the identity map of $R$ for
each $i=1,\dots,n$, (we write $\sigma_{i}={\rm id}_{R}$), we say that $A$ is a
skew PBW extension of derivation type. Similarly, if $\delta_{i}\in\Delta$ is
zero, for every $i$, 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:

  * •

[51, Theorem 2.3] If $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ is a
quasi-commutative skew PBW extension over a ring $R$, then $A$ is isomorphic
to an iterated skew polynomial ring of endomorphism type.

  * •

[51, Example 5(3)] If
$R[x_{1};\sigma_{1},\delta_{1}]\dotsb[x_{n};\sigma_{n},\delta_{n}]$ is an
iterated skew polynomial ring such that $\sigma_{i}$ is injective, for $1\leq
i\leq n$; $\sigma_{i}(r)$, $\delta_{i}(r)\in R$, for $r\in R$ and $1\leq i\leq
n$; $\sigma_{j}(x_{i})=cx_{i}+d$, for $i<j$, and $c,d\in R$, where $c$ has a
left inverse; $\delta_{j}(x_{i})\in R+Rx_{1}+\dotsb+Rx_{n}$, for $i<j$, then
$R[x_{1};\sigma_{1},\delta_{1}]\dotsb[x_{n};\sigma_{n},\delta_{n}]\cong\sigma(R)\langle
x_{1},\dotsc,x_{n}\rangle$. In general, skew polynomial rings of injective
type are strictly contained in skew PBW extensions. This fact is not possible
for PBW extensions. For instance, the quantum plane $\Bbbk\langle
x,y\rangle/\langle xy-qyx\mid q\in\Bbbk^{*}\rangle$ is an Ore extension of
injective type given by $\Bbbk[y][x;\sigma]$, where $\sigma(y)=qy$, but cannot
be expressed as a PBW extension.

  * •

[68, Remark 2.4 (ii)] Skew PBW extensions of endomorphism type are more
general than iterated skew polynomial rings of endomorphism type. With the aim
of illustrating the differences between these structures, we consider the
situations with two and three indeterminates. If we take the iterated skew
polynomial ring of endomorphism type given by $R[x;\sigma_{x}][y;\sigma_{y}]$,
by definition we have the following relations: $xr=\sigma_{x}(r)x$,
$yr=\sigma_{y}(r)y$, and $yx=\sigma_{y}(x)y$, for any element $r\in R$. On the
other hand, if we have $\sigma(R)\langle x,y\rangle$ a skew PBW extension of
endomorphism type over $R$, then for any $r\in R$ we have the relations
$xr=\sigma_{1}(r)x$, $yr=\sigma_{2}(r)y$, 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 $R$. When we compare the defining relations of both
algebraic structures, it is clear which one of them is more general.

Now, 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 $\sigma(R)\langle x,y,z\rangle$, we have
the relations given by $xr=\sigma_{1}(r)x$, $yr=\sigma_{2}(r)y$,
$zr=\sigma_{3}(r)z$, $yx=d_{1,2}xy+r_{0}+r_{1}x+r_{2}y+r_{3}z$,
$zx=d_{1,3}xz+r_{0}^{\prime}+r_{1}^{\prime}x+r_{2}^{\prime}y+r_{3}^{\prime}z$,
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 $R$. Again, one can see the
generality of the skew PBW extensions. Of course, as the number of
indeterminates increases, the differences between both algebraic structures
are more remarkable.

######  Example 2.5.

A great variety of algebras can be expressed as skew PBW extensions.
Enveloping algebras of finite dimensional Lie algebras, families of
differential operators rings, Weyl algebras, Ore extensions of injective type,
some types of Auslander-Gorenstein rings, some skew Calabi Yau algebras,
Artin-Schelter regular algebras, Koszul algebras, examples of quantum
polynomials, some quantum universal enveloping algebras, and many other
algebras of great interest in noncommutative algebraic geometry and
noncommutative differential geometry illustrate the generality of skew PBW
extensions. Details of these examples and others can be found in [16, 18, 67].

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

######  Proposition 2.6.

([17, 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 $p_{\alpha,\beta}\neq 0$.

We need to establish a criterion which allows us to extend the family $\Sigma$
of injective endomorphisms and the family of $\Sigma$-derivations $\Delta$ of
the ring $R$, to any skew PBW extension $A$ over $R$ (c.f. Artamonov [5] and
Venegas [70] who presented a study of derivations and automorphisms of skew
PBW extensions, respectively). With this aim, for the next result consider the
pair $(\Sigma,\Delta)$, the system of endomorphisms and $\Sigma$-derivations
of $R$ with respect to $A$.

######  Proposition 2.7.

([61, Theorem 5.1]) Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a
skew PBW extension over a ring $R$. Suppose that
$\sigma_{i}\delta_{j}=\delta_{j}\sigma_{i},\
\delta_{i}\delta_{j}=\delta_{j}\delta_{i}$, and
$\delta_{k}(d_{i,j})=\delta_{k}(r_{l}^{(i,j)})=0$, for $1\leq i,j,k,l\leq n$,
where $d_{i,j}$ and $r_{l}^{(i,j)}$ are the elements established in Definition
2.1. Consider an element $f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}\in A$. If
$\overline{\sigma_{k}}:A\to A$ and $\overline{\delta_{k}}:A\to A$ are the
functions given by
$\overline{\sigma_{k}}(f):=\sigma_{k}(a_{0})+\sigma_{k}(a_{1})X_{1}+\dotsb+\sigma_{k}(a_{m})X_{m}$
and
$\overline{\delta_{k}}(f):=\delta_{k}(a_{0})+\delta_{k}(a_{1})X_{1}+\dotsb+\delta_{k}(a_{m})X_{m}$,
respectively, and $\overline{\sigma_{k}}(r):=\sigma_{k}(r)$, for every $1\leq
k\leq n$ and $r\in R$, then $\overline{\sigma_{k}}$ is an injective
endomorphism of $A$ and $\overline{\delta_{k}}$ is a
$\overline{\sigma_{k}}$-derivation of $A$, for each $k$.

As we said in Section 1, Krempa [39] introduced the notion of rigidness as a
generalization of reduced rings. Now, if $R$ is a ring and
$\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$ is a family of endomorphisms of
$R$, then $\Sigma$ is called a _rigid endomorphisms family_ if
$a\sigma^{\alpha}(a)=0$ implies $a=0$, where $a\in R$ and
$\alpha\in\mathbb{N}^{n}$. $R$ is called $\Sigma$-_rigid_ if there exists a
rigid endomorphisms family $\Sigma$ of $R$ [61], Definition 3.1. Following
Annin [3] or Hashemi and Moussavi [25], if $R$ is a ring, $\sigma$ is an
endomorphism of $R$, and $\delta$ is a $\sigma$-derivation of $R$, then $R$ is
said to be $\sigma$-compatible if for each $a,b\in R$, $ab=0$ if and only if
$a\sigma(b)=0$; in this case is said that $\sigma$ is compatible (necessarily,
the endomorphism $\sigma$ is injective). $R$ is called $\delta$-compatible if
for each $a,b\in R$, $ab=0$ implies $a\delta(b)=0$. If $R$ is both
$\sigma$-compatible and $\delta$-compatible, $R$ is called
($\sigma,\delta$)-compatible. The corresponding notion of compatibility, and
an even weaker notion, have been formulated for skew PBW extensions as the
following definition shows.

######  Definition 2.8.

Consider a ring $R$ with finite families of endomorphisms $\Sigma$ and
$\Sigma$-derivations $\Delta$, respectively.

  1. (1)

([22, Definition 3.1]; [62, Definition 3.2]) $R$ is said to be
$\Sigma$-compatible if for each $a,b\in R$, $a\sigma^{\alpha}(b)=0$ if and
only if $ab=0$, where $\alpha\in\mathbb{N}^{n}$. In this case, we say that
$\Sigma$ is a compatible family of endomorphisms of $R$. $R$ is said to be
$\Delta$-compatible if for each $a,b\in R$, it follows that $ab=0$ implies
$a\delta^{\beta}(b)=0$, where $\beta\in\mathbb{N}^{n}$. If $R$ is both
$\Sigma$-compatible and $\Delta$-compatible, then $R$ is called
$(\Sigma,\Delta)$-compatible.

  2. (2)

[64, Definition 4.1] $R$ is said to be weak $\Sigma$-compatible if for each
$a,b\in R$, $a\sigma^{\alpha}(b)\in N(R)$ if and only if $ab\in N(R)$, where
$\alpha\in\mathbb{N}^{n}$. $R$ is said to be weak $\Delta$-compatible if for
each $a,b\in R$, $ab\in N(R)$ implies $a\delta^{\beta}(b)\in N(R)$, where
$\beta\in\mathbb{N}^{n}$. If $R$ is both weak $\Sigma$-compatible and weak
$\Delta$-compatible, then $R$ is called weak $(\Sigma,\Delta)$-compatible.

Ring-theoretical properties of skew PBW extensions over compatible rings have
been investigated by the authors [63, 65].

######  Remark 2.9.

It is straightforward to prove that if $\Sigma$ is a finite family of
endomorphisms of a ring $R$, then $R$ is $\Sigma$-compatible if and only if
$R$ is $\sigma_{i}$-compatible, for every $1\leq i\leq n$.

######  Lemma 2.10.

([22, Lemma 3.3]; [62, Proposition 3.8]) Let $R$ be a
$(\Sigma,\Delta)$-compatible ring. For every $a,b\in R$, we have the
following:

  1. (1)

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

  2. (2)

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

  3. (3)

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

######  Lemma 2.11.

If $R$ is a $\Sigma$-compatible ring, $a,b\in R$ and
$\alpha,\beta\in\mathbb{N}^{n}$, then the following assertions are equivalent:

  1. (1)

$ab\in N(R)$.

  2. (2)

$a\sigma^{\alpha}(b)\in N(R)$.

  3. (3)

$\sigma^{\alpha}(b)a\in N(R)$.

  4. (4)

$\sigma^{\alpha}(a)b\in N(R)$.

  5. (5)

$b\sigma^{\alpha}(a)\in N(R)$.

  6. (6)

$\sigma^{\alpha}(a)\sigma^{\beta}(b)\in N(R)$.

###### Proof.

Let $R$ be a $\Sigma$-compatible ring, $a,b\in R$ and
$\alpha,\beta\in\mathbb{N}^{n}$.

(1) $\Leftrightarrow$ (2) By definition, $ab\in
N(R)\Leftrightarrow(ab)^{k}=0$, for some positive integer $k$. If $k=1$,
$ab=0$ if and only if $a\sigma^{\alpha}(b)$. If $k=2$, $0=abab\Leftrightarrow
0=a\sigma^{\alpha}(bab)=a\sigma^{\alpha}(b)\sigma^{\alpha}(ab)\Leftrightarrow
a\sigma^{\alpha}(b)ab=0\Leftrightarrow
0=a\sigma^{\alpha}(b)a\sigma^{\alpha}(b)=(a\sigma^{\alpha}(b))^{2}$, where all
equivalences are due to the $\Sigma$-compatibility of $R$. Continuing with
this process, $(ab)^{k}=0\Leftrightarrow(a\sigma^{\alpha}(b))^{k}=0$ Thus,
$ab\in N(R)$ if and only if $a\sigma^{\alpha}(b)\in N(R)$.

(2) $\Leftrightarrow$ (3) It is clear.

(1) $\Leftrightarrow$ (4) Again, since $ab\in N(R)\Leftrightarrow(ab)^{k}=0$,
for some positive integer $k$, if $k=1$, Lemma 2.10 implies that
$ab=0\Leftrightarrow\sigma^{\alpha}(a)b=0$. If $k=2$, $0=abab\Leftrightarrow
0=\sigma^{\alpha}(a)bab\Leftrightarrow\sigma^{\alpha}(a)b\sigma^{\alpha}(ab)=\sigma^{\alpha}(a)b\sigma^{\alpha}(a)\sigma^{\alpha}(b)$,
$\Leftrightarrow\sigma^{\alpha}(a)b\sigma^{\alpha}(a)b=0$
$\Leftrightarrow(\sigma^{\alpha}(a)b)^{2}=0$, where the first equivalence is
due to Lemma 2.10, and the others equivalences are due to the
$\Sigma$-compatibility of $R$. Continuing in this way, we can see that
$(ab)^{k}=0\Leftrightarrow(\sigma^{\alpha}(a)b)^{k}=0$. Hence $ab\in
N(R)\Leftrightarrow\sigma^{\alpha}(a)b\in N(R)$.

(4) $\Leftrightarrow$ (5) It is immediate.

(5) $\Leftrightarrow$ (6) It follows from (1) $\Leftrightarrow$ (2) by
replacing $a$ by $\sigma^{\alpha}(a)$ and $\sigma^{\alpha}(b)$ by
$\sigma^{\beta}(b)$. ∎

To finish this section, we recall the notion of Armendariz ring in the setting
of commutative and noncommutative rings. Following Rege and Chhawchharia [60],
a ring $R$ is called Armendariz if for elements
$f(x)=\sum_{i=0}^{m}a_{i}x^{i},\ g(x)=\sum_{j=0}^{l}b_{j}x^{j}\in R[x]$ (where
$R[x]$ is the commutative polynomial ring with an indeterminate $x$ over $R$),
$f(x)g(x)=0$ implies $a_{i}b_{j}=0$, for all $i,j$. The importance of this
notion lies in its natural and useful role in understanding the relation
between the annihilators of the ring $R$ and the annihilators of the
polynomial ring $R[x]$. For example, Armendariz [4, Lemma 1], showed that a
reduced ring always satisfies this condition. For skew polynomial rings, the
notion of Armendariz has been also studied by several authors (see [1, 4, 29,
31, 37, 44, 60]). Hirano [27] generalized this notion in the following way: a
ring $R$ is called quasi-Armendariz if for $f(x)=\sum_{i=0}^{m}a_{i}x^{i},\
g(x)=\sum_{j=0}^{l}b_{j}x^{j}$ in $R[x]$, $f(x)R[x]g(x)=0$ implies
$a_{i}Rb_{j}=0$, for all $i,j$. It is well known that reduced rings are
Armendariz, and Armendariz rings are quasi-Armendariz, but the converse are
not true in general [13, Proposition 2.1].

In the noncommutative setting, Hashemi et al. [25] investigated a
generalization of $\sigma$-rigid rings by introducing the (SQA1) condition
which is a version of quasi-Armendariz rings for skew polynomial rings. Let
$R[x;\sigma,\delta]$ be an Ore extension over $R$. $R$ is said to satisfy the
(SQA1) condition if whenever $f(x)R[x;\sigma,\delta]g(x)=0$ for
$f(x)=\sum_{i=0}^{m}a_{i}x^{i}$ and $g(x)=\sum_{j=0}^{m}b_{j}x^{j}\in
R[x;\sigma,\delta]$, then $a_{i}Rb_{j}=0$ for all $i,j$. Later, Reyes and
Suárez [62] considered this condition for skew PBW extensions. For
$A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ a skew PBW extension over $R$,
$R$ is said to satisfy the (SQA1) condition if whenever $fAg=0$ for
$f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}$ and
$g=b_{0}+b_{1}Y_{1}+\dotsb+b_{t}Y_{t}$ elements of $A$, then $a_{i}Rb_{j}=0$,
for every $i,j$. Relationships between the notions of compatibility, (SQA1),
and Armendariz rings, in the context of skew PBW extensions, can be found in
[61, 65]. All of them will be useful in the next sections.

##  3\. $\Sigma-$skew reflexive rings

We present the original results of the paper on $\Sigma$-skew reflexive rings.
We start with the following definition which establishes the generalizations
of skew reflexive endomorphism and $\sigma$-skew reflexive ring mentioned in
the Introduction.

######  Definition 3.1.

Let $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$ be a finite family of
endomorphisms of a ring $R$. $\Sigma$ is called _right_ (resp., _left_) skew
reflexive if for $a,b\in R$, $aRb=0$ implies $bR\sigma^{\alpha}(a)=0$ (resp.,
$\sigma^{\alpha}(b)Ra=0$), where $\alpha\in\mathbb{N}^{n}\ \backslash\
\\{0\\}$. $R$ is called right (resp., left) $\Sigma$-skew reflexive if there
exists a right (resp., left) skew reflexive family of endomorphisms $\Sigma$
of $R$. If $R$ is both right and left $\Sigma$-skew reflexive, then $R$ is
said to be $\Sigma$-skew reflexive.

Reflexive rings are $\Sigma$-skew reflexive. Since reduced rings and
reversible rings are reflexive, then both rings are $\Sigma$-skew reflexive.
Right (resp., left) $\sigma$-skew reflexive rings are right (resp., left)
$\Sigma$-skew reflexive.

######  Example 3.2.

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

Consider the ring $S_{2}(\mathbb{Z})$ given by

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

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

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

Consider $ARB=0$, for all $R\in S_{2}(\mathbb{Z})$, where

$A=\begin{pmatrix}a&a^{\prime}\\\ 0&a\end{pmatrix},\ \
B=\begin{pmatrix}b&b^{\prime}\\\ 0&b\end{pmatrix}\ \ \text{and}\ \
R=\begin{pmatrix}r&r^{\prime}\\\ 0&r\end{pmatrix}$.

We have $arb=0$ and $arb^{\prime}+ar^{\prime}b+bra^{\prime}=0$, for every
$r\in\mathbb{Z}$, whence $a=0$ or $b=0$. If $a=0$, then $bra^{\prime}=0$, for
every $r\in\mathbb{Z}$, hence $b=0$ or $a^{\prime}=0$. If $b=0$, then
$arb^{\prime}=0$, for each $r\in\mathbb{Z}$, hence $a=0$ or $b^{\prime}=0$.
First, we consider the case $BR\sigma_{1}(A)=BRA$:

$BR\sigma_{1}(A)=\begin{pmatrix}bra&bra^{\prime}+br^{\prime}a+b^{\prime}ra\\\
0&bra\end{pmatrix}$.

From the previous observation, it follows that $BR\sigma_{1}(A)=0$. Now,
consider the case $BR\sigma_{2}(A)$. A calculation shows us that

$BR\sigma_{2}(A)=\begin{pmatrix}bra&-bra^{\prime}+br^{\prime}a+b^{\prime}ra\\\
0&bra\end{pmatrix}$.

Again, making use of the initial observation, we can see that
$BR\sigma_{2}(A)=0$. $S_{2}(\mathbb{Z})$ is right $\sigma_{2}$-skew reflexive
[42, Proposition 2.11]. Finally, we present the case $BR\sigma_{3}(A)$:

$BR\sigma_{3}(A)=\begin{pmatrix}bra&br^{\prime}a+b^{\prime}ra\\\
0&bra\end{pmatrix}$.

We obtain that $bra=0$, since $BRA=0$. If $a=0$, then
$br^{\prime}a+b^{\prime}ra=0$, for every $r,r^{\prime}\in\mathbb{Z}$. If
$b=0$, it follows that
$b^{\prime}ra+br^{\prime}a+bra^{\prime}=br^{\prime}a+b^{\prime}ra=0$, for each
$r,r^{\prime}\in\mathbb{Z}$. This implies that $BR\sigma_{3}(A)=0$. We can
also observe that
$\sigma_{2}\circ\sigma_{3}=\sigma_{3}\circ\sigma_{2}=\sigma_{3}$, which
implies that $S_{2}(\mathbb{Z})$ is right $\Sigma$-skew reflexive with
$\Sigma=\left\\{\sigma_{1},\sigma_{2},\sigma_{3}\right\\}$.

Since $\sigma_{1}$ is the identity endomorphism, then we have that
$S_{2}(\mathbb{Z})$ is a reflexive ring. Now, $S_{2}(\mathbb{Z})$ is not
$\sigma_{3}$-compatible since for

$A=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\ \ B=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},$

$A\sigma_{3}(B)=0$ but $AB=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\neq 0.$
According to Remark 2.9, $S_{2}(\mathbb{Z})$ is not $\Sigma$-compatible.

The following result shows that under conditions of compatibility, the
composition of left (resp. right) skew reflexive endomorphisms is left (resp.
right) skew reflexive endomorphism.

######  Proposition 3.3.

Composition of compatible left (resp. right) skew reflexive endomorphisms is
left (resp. right) skew reflexive.

###### Proof.

Let $\sigma_{1}$ and $\sigma_{2}$ be compatible endomorphisms of a ring $R$
and consider $a,b\in R$. Suppose that $aRb=0$, i.e. $arb=0$, for every $r\in
R$. If $\sigma_{1}$ and $\sigma_{2}$ are left skew reflexive, then
$\sigma_{1}(b)ra=\sigma_{2}(b)ra=0$. Hence
$\sigma_{1}(\sigma_{2}(b)ra)=\sigma_{1}(\sigma_{2}(b))\sigma_{1}(ra)=\sigma_{1}(\sigma_{2}(b))ra=0$,
since $\sigma_{1}$ is compatible. So, $\sigma_{1}\circ\sigma_{2}$ is left skew
reflexive. Now, if $\sigma_{1}$ and $\sigma_{2}$ are right skew reflexive
endomorphisms, then $arb=0$ implies $br\sigma_{1}(a)=br\sigma_{2}(a)=0$,
whence
$\sigma_{1}(br\sigma_{2}(a))=\sigma_{1}(br)\sigma_{1}(\sigma_{2}(a))=0$. Since
$\sigma_{1}$ is compatible, $\sigma_{1}(br)\sigma_{1}(\sigma_{2}(a))=0$
implies $br\sigma_{1}(\sigma_{2}(a))=0$, and therefore
$\sigma_{1}\circ\sigma_{2}$ is a right skew reflexive endomorphism. ∎

From Proposition 3.3 we deduce the following results.

######  Corollary 3.4.

Composition of a finite family of compatible left (resp. right) skew reflexive
endomorphisms is a left (resp. right) skew reflexive endomorphism. If $R$ is
$\sigma$-compatible then $R$ is right (resp., left) $\sigma$-skew reflexive if
and only if it is right (resp., left) $\sigma^{k}$-skew reflexive, for all
$k\geq 1$. Thus, if $R$ is $\sigma$-compatible then $R$ is $\sigma$-skew
reflexive if and only if it is $\sigma^{k}$-skew reflexive, for all $k\geq 1$.

######  Proposition 3.5.

Let $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$ be a compatible family of
endomorphisms of $R$. Then $\Sigma$ is a right (left) skew reflexive family if
and only if each $\sigma_{k}\in\Sigma$ is a right (left) skew reflexive
endomorphism.

###### Proof.

Let $\sigma_{k}\in\Sigma$. Note that for
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$,
$\sigma^{\alpha}=\sigma_{k}$, for $\alpha_{i}=0$ if $i\neq k$ and
$\alpha_{k}=1$. If $\Sigma$ is a right (resp., left) skew reflexive family of
endomorphisms, then for $a,b\in R$, $aRb=0$ implies
$bR\sigma^{\alpha}(a)=0=bR\sigma_{k}(a)$ (resp.,
$\sigma^{\alpha}(b)Ra=\sigma_{k}(b)Ra=0$). So, $\sigma_{k}$ is a right (resp.,
left) skew reflexive endomorphism. On the other hand, if each
$\sigma_{k}\in\Sigma$ is a compatible right (left) skew reflexive endomorphism
then by Corollary 3.4 we have that $\sigma^{\alpha}$ is right (resp., left)
skew reflexive, for every $\alpha\in\mathbb{N}^{n}$. Then for $a,b\in R$,
$aRb=0$ implies $bR\sigma^{\alpha}(a)$ (resp., $\sigma^{\alpha}(b)Ra=0$), for
every $\alpha\in\mathbb{N}^{n}$. Therefore $\Sigma$ is a right (left) skew
reflexive family. ∎

######  Proposition 3.6.

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a quasi-commutative skew
PBW extension over a $\Sigma$-skew reflexive ring $R$, that is, $A$ is an
iterated skew polynomial ring of injective endomorphism type over $R$. If $R$
satisfies the (SQA1) condition, then $A$ is reflexive.

###### Proof.

Let $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$ be the family of injective
endomorphisms of $R$ as in Proposition 2.2. Consider
$f=a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m}$ and
$g=b_{0}+b_{1}Y_{1}+\dotsb+b_{t}Y_{t}$ elements of $A$. If $fAg=0$, by the
(SQA1) condition of $R$ we obtain $a_{i}Rb_{j}=0$, for each $i,j$. Since $R$
is $\Sigma$-skew reflexive,
$b_{j}R\sigma^{\alpha}(a_{i})=\sigma^{\alpha}(b_{j})Ra_{i}=0$, where
$\alpha\in\mathbb{N}^{n}\ \backslash\ \\{0\\}$, and each $1\leq i\leq m$,
$1\leq j\leq t$. As $A$ is quasi-commutative, by Proposition 2.6 we know that
for every $x^{\alpha}\in{\rm Mon}(A)$ and every $0\neq r\in R$, there exists
an element $\sigma^{\alpha}(r)\in R\ \backslash\ \\{0\\}$ such that
$x^{\alpha}r=\sigma^{\alpha}(r)x^{\alpha}$, and for every
$x^{\alpha},x^{\beta}\in{\rm Mon}(A)$, there exists an element
$d_{\alpha,\beta}\in R$ such that
$x^{\alpha}x^{\beta}=d_{\alpha,\beta}x^{\alpha+\beta}$, where
$d_{\alpha,\beta}$ is left invertible. Thus, applying these commutation rules
to the product
$(b_{0}+b_{1}Y_{1}+\dotsb+b_{t}Y_{t})(c_{0}+c_{1}Z_{1}+\dotsb+c_{l}Z_{l})(a_{0}+a_{1}X_{1}+\dotsb+a_{m}X_{m})$,
where $c_{0}+c_{1}Z_{1}+\dotsb+c_{l}Z_{l}$ is an arbitrary element of $A$, and
taking into account that
$b_{j}R\sigma^{\alpha}(a_{i})=\sigma^{\alpha}(b_{j})Ra_{i}=0$, for
$\alpha\in\mathbb{N}^{n}\ \backslash\ \\{0\\}$, and every $i$, $j$, then
$gAf=0$, that is, $A$ is reflexive. ∎

From Proposition 3.6 we obtain that if $R$ is a reflexive ring, then $R[x]$ is
right idempotent reflexive [41, Theorem 3.13 (1)].

######  Proposition 3.7.

$\Sigma$-compatible semiprime rings are $\Sigma$-skew reflexive.

###### Proof.

Suppose that $R$ is semiprime and $\Sigma$-compatible for
$\Sigma=\\{\sigma_{1},\dotsc,\sigma_{n}\\}$ a finite family of endomorphisms
of $R$. Let $a,b\in R$ such that $aRb=0$. By the $\Sigma$-compatibility of
$R$, $aR\sigma^{\alpha}(b)=0$, where $\alpha\in\mathbb{N}^{n}$, whence
$\sigma^{\alpha}(b)RaR\sigma^{\alpha}(b)Ra=0$. Since $R$ is semiprime,
$\sigma^{\alpha}(b)Ra=0$, and thus $R$ is left $\Sigma$-skew reflexive. By
Remark 2.9, $aRb=0$ implies $\sigma^{\alpha}(a)Rb=0$, and so
$bR\sigma^{\alpha}(a)RbR\sigma^{\alpha}(a)=0$. The semiprimeness of $R$
implies $bR\sigma^{\alpha}(a)=0$, and therefore $R$ is right $\Sigma$-skew
reflexive. ∎

Propositions 3.6 and 3.7 guarantee that quasi-commutative skew PBW extensions
over $\Sigma$-compatible semiprime rings satisfying the (SQA1) condition are
reflexive.

######  Remark 3.8.

Following Birkenmeier et al. [11], a ring $R$ is called right (left)
principally quasi-Baer (or simply, right (left) p.q.-Baer) ring if the right
annihilator of a principal right ideal of $R$ is generated by an idempotent.
$R$ is said to be a p.q.-Baer if it is both right and left p.q.-Baer. Having
in mind that Baer properties have been studied in the setting of skew PBW
extensions [61, 62], if $A$ is a skew PBW extension right p.q.-Baer ring, then
$A$ is a semiprime ring if and only if $A$ is a reflexive ring [41,
Proposition 3.15].

##  4\. Skew PBW extension over $\Sigma-$skew RNP rings

Motivated by the notion put forward by Bhattacharjee [9, Definition 1.1], we
introduce the following definition.

######  Definition 4.1.

Let $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$ be a finite family of
endomorphisms of a ring $R$. $\Sigma$ is called a _right_ (resp., _left_) skew
RNP family if for $a,b\in N(R)$, $aRb=0$ implies $bR\sigma^{\alpha}(a)=0$
(resp., $\sigma^{\alpha}(b)Ra=0$), for every nonzero element
$\alpha\in\mathbb{N}^{n}$. $R$ is said to be right (resp., left) $\Sigma$-skew
RNP if there exists a right (resp., left) skew RNP family of endomorphism
$\Sigma$ of $R$. If $R$ is both right and left $\Sigma$-skew RNP, then we say
that $R$ is $\Sigma$-skew RNP.

It is straightforward to show that right (resp., left) $\Sigma$-skew reflexive
rings are right (resp., left) $\Sigma$-skew RNP, reduced rings are
$\Sigma$-skew RNP, for any finite family of endomorphisms $\Sigma$, and RNP
rings are $\Sigma$-skew RNP considering $\Sigma$ as the singleton consisting
of the identity function of $R$.

######  Remark 4.2.

Note that for $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$,
$\sigma^{\alpha}(N(R))\subseteq N(R)$, for each $\alpha\in\mathbb{N}^{n}$.
Indeed, if $r\in N(R)$ then $r^{m}=0$, for some $m\geq 1$. So,
$\sigma^{\alpha}(r)\in\sigma^{\alpha}(N(R))$ implies that
$\sigma^{\alpha}(r^{n})=(\sigma^{\alpha}(r))^{n}=0$, whence
$\sigma^{\alpha}(r)\in N(R)$.

######  Example 4.3.

Considering Example 3.2. It is clear that $S_{2}(\mathbb{Z})$ is a right
$\Sigma$-skew RNP ring with
$\Sigma=\left\\{\sigma_{1},\sigma_{2},\sigma_{3}\right\\}$. Furthermore, we
can see that $N(S_{2}(\mathbb{Z}))=\left\\{\begin{pmatrix}0&b\\\
0&0\end{pmatrix}\mid b\in\mathbb{Z}\right\\}$.

######  Example 4.4.

Let $\mathbb{F}_{4}=\left\\{0,1,a,a^{2}\right\\}$ be the finite field of four
elements. Consider the ring of polynomials $\mathbb{F}_{4}[z]$ and let
$R=\frac{\mathbb{F}_{4}[z]}{\left\langle z^{2}\right\rangle}$. For simplicity,
we identify the elements of $\mathbb{F}_{4}[z]$ with their images in $R$. Let
$\Sigma=\left\\{\sigma_{i,j}\right\\}$ be the family of homomorphisms of $R$
defined by $\sigma_{i,j}(a)=a^{i}$ and $\sigma_{i,j}(z)=a^{j}z$, for all
$a\in\mathbb{Z}_{3}$ with $1\leq i\leq 2$ and $0\leq j\leq 2$. Let us consider
the skew PBW extension defined as $A=\sigma(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$, under the
following commutation relations:
$x_{i,j}x_{i^{\prime},j^{\prime}}=x_{i^{\prime},j^{\prime}}x_{i,j}$, for all
$1\leq i,i^{\prime}\leq 2$ and $0\leq j,j^{\prime}\leq 2$. On the other hand,
for $a^{r}z\in R$,
$x_{i,j}a^{r}z=\sigma_{i,j}(a^{r}z)x_{i,j}=(a^{r})^{i}a^{j}zx_{i,j}=a^{ri+j}zx_{i,j}$,
where $a^{ri+j}\in\mathbb{F}_{4}$, $1\leq i\leq 2$ and $1\leq r,j\leq 2$. Of
course, this example can be extended to any finite field $\mathbb{F}_{p^{n}}$
with $p$ a prime number.

Let us show that $R$ is a $\Sigma$-skew RNP ring. Note first that
$\sigma_{1,0}$ is the identity homomorphism over $R$ and $\Sigma$ is closed
under composition, that is, $\sigma_{i,j}^{\alpha}\in\Sigma$, for all
$\alpha\in\mathbb{N}^{6}$. Notice that the set of nilpotent elements of $R$ is
the ideal generated by $z$, that is, $N(R)=\left\langle z\right\rangle$. Let
$f,g\in N(R)$ such that $fRg=0$. Notice that $f=a^{r}z$ and $g=a^{s}z$, for
some $0\leq r,s\leq 2$. Futhermore, $\sigma_{i,j}^{\alpha}(f)=a^{k}z$, for all
$\alpha\in\mathbb{N}^{6}$ and some $0\leq k\leq 2$. Hence, we have
$gR\sigma_{i,j}^{\alpha}(f)=0$, for all $\alpha\in\mathbb{N}^{6}$ with $1\leq
i\leq 2$ and $0\leq j\leq 2$. In this way, $R$ is a right $\Sigma$-skew RNP
ring.

Due to [9, Proposition 1.8], $R$ is $\sigma$-skew RNP if and only if it is
$\sigma^{2k-1}$-skew RNP, for all $k\geq 1$. Note that if $R$ is
$\sigma$-compatible then $R$ is $\sigma$-skew RNP if and only if it is
$\sigma^{k}$-skew RNP for all $k\geq 1$, as it is shown below. Since
$\sigma(N(R))\subseteq N(R)$ for every endomorphism $\sigma$ of $R$ and since
right (resp., left) $\sigma$-skew reflexive rings are right (resp., left)
$\sigma$-skew RNP, Proposition 3.3 implies immediately the following:

######  Corollary 4.5.

Let $\sigma$ be an endomorphism of $R$.

  1. (1)

Composition of finite family of compatible left (resp. right) skew NRP
endomorphisms is a left (resp. right) skew NRP endomorphism.

  2. (2)

If $R$ is $\sigma$-compatible then $R$ is right (resp., left) $\sigma$-skew
NRP if and only if it is right (resp., left) $\sigma^{k}$-skew NRP, for all
$k\geq 1$. Thus, if $R$ is $\sigma$-compatible then $R$ is $\sigma$-skew NRP
if and only if it is $\sigma^{k}$-skew NRP, for all $k\geq 1$.

######  Proposition 4.6.

If $R$ is a $\Sigma$-compatible ring, then $R$ is left (resp., right)
$\Sigma$-skew RNP if and only if it is left (resp., right) $\sigma_{k}$-skew
RNP for each $\sigma_{k}\in\Sigma$.

###### Proof.

The proof is similar to that of Proposition 3.5, but using Remark 4.2 and
Corollary 4.5. ∎

######  Proposition 4.7.

If $R$ is a $\Sigma$-compatible ring, then the following assertions are
equivalent:

  1. (1)

$R$ is RNP.

  2. (2)

$R$ is right $\Sigma$-skew RNP.

  3. (3)

$R$ is left $\Sigma$-skew RNP.

###### Proof.

(1) $\Rightarrow$ (2) Suppose that $R$ is RNP and consider $a,b\in N(R)$ such
that $aRb=0$, that is, $arb=0$, for any $r\in R$. By the
$\Sigma$-compatibility of $R$, $\sigma^{\alpha}(a)rb=0$, i.e.
$\sigma^{\alpha}(a)Rb=0$ (Lemma 2.10). Since $a\in N(R)$,
$\sigma^{\alpha}(a)\in\sigma^{\alpha}(N(R))$, and by Remark 4.2, we have
$\sigma^{\alpha}(a)\in N(R)$. As $R$ is RNP, $\sigma^{\alpha}(a)Rb=0$ implies
that $bR\sigma^{\alpha}(a)=0$, whence $R$ is right $\Sigma$-skew RNP.

(2) $\Rightarrow$ (1) Let $R$ be right $\Sigma$-skew RNP and let $a,b\in N(R)$
such that $aRb=0$, i.e. $arb=0$, for any $r\in R$. Since $R$ is
$\Sigma$-compatible, $ar\sigma^{\alpha}(b)=0$, for every
$\alpha\in\mathbb{N}^{n}$. Using that $b\in N(R)$, we obtain
$\sigma^{\alpha}(b)\in\sigma^{\alpha}(N(R))$, and by Remark 4.2,
$\sigma^{\alpha}(b)\in N(R)$. Now, as $R$ is right $\Sigma$-skew RNP, then
$\sigma^{\alpha}(b)R\sigma^{\alpha}(a)=0$. So, for each $r\in R$,
$\sigma^{\alpha}(bra)=\sigma^{\alpha}(b)\sigma^{\alpha}(r)\sigma^{\alpha}(a)=0$.
Since $\Sigma$ is compatible, then $\sigma^{\alpha}$ is injective, and so
$bra=0$, for all $r\in R$. Therefore $bRa=0$, and hence $R$ is RNP.

With a reasoning similar to the previous one, one can prove (1)
$\Leftrightarrow$ (3). ∎

######  Corollary 4.8.

([9, Proposition 1.7]) Let $R$ be a $\sigma$-compatible ring. Then the
following are equivalent: (1) $R$ is RNP (2) $R$ is right $\sigma$-skew RNP
(3) $R$ is left $\sigma$-skew RNP.

######  Corollary 4.9.

If $R$ is a $\Sigma$-compatible ring, then the following are equivalent:

  1. (1)

$R$ is reflexive.

  2. (2)

$R$ is right $\Sigma$-skew reflexive.

  3. (3)

$R$ is left $\Sigma$-skew reflexive.

######  Remark 4.10.

It is straightforward to prove that nil-reversible rings are RNP.

The following property relates nil-reversible and $\Sigma$-compatible rings
with $\Sigma$-skew RNP rings.

######  Proposition 4.11.

If $R$ is nil-reversible and $\Sigma$-compatible then $R$ is $\Sigma$-skew
RNP.

###### Proof.

Let $\Sigma=\\{\sigma_{1},\dots,\sigma_{n}\\}$ be a finite family of
endomorphisms of a nil-reversible and $\Sigma$-compatible ring $R$. Suppose
that $aRb=0$, for $a,b\in N(R)$. By Remark 4.10, $bRa=0$, and so
$bR\sigma^{\alpha}(a)=0$, since $R$ is $\Sigma$-compatible. Thus $R$ is right
$\Sigma$-skew RNP. From Lemma 2.10, we have that $bRa=0$ implies
$\sigma^{\alpha}(b)Ra=0$, and therefore $R$ is left $\Sigma$-skew RNP. ∎

Due to [16, Proposition 3.2.1], a skew PBW extension $A$ over a domain $R$ is
also a domain, and so $N(A)=0$. In this way, if $A$ is a skew PBW extension
over a domain $R$ which satisfies the conditions established in Proposition
2.7, then $A$ is $\overline{\Sigma}$-skew RNP for
$\overline{\Sigma}=\\{\overline{\sigma_{1}},\dots,\overline{\sigma_{n}}\\}$,
where $\overline{\sigma_{k}}$ is as in Proposition 2.7.

######  Proposition 4.12.

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

  1. (1)

Both $R$ and $A$ are reflexive.

  2. (2)

$R$ is $\Sigma$-skew RNP.

  3. (3)

If the conditions established in Proposition 2.7 hold, then $A$ is
$\overline{\Sigma}$-skew RNP.

###### Proof.

(1) By [61, Theorem 4.4], we have that both $R$ and $A$ is reduced. Since
reduced rings are reflexive, then $R$ and $A$ are reflexive.

(2) As $R$ is reduced, $R$ is $\Sigma$-skew RNP for any finite family of
endomorphisms $\Sigma$.

(3) Since $A$ is reduced, it follows that $A$ is $\Sigma$-skew RNP for any
finite family of endomorphisms. In particular, $A$ is $\overline{\Sigma}$-skew
RNP for
$\overline{\Sigma}=\\{\overline{\sigma_{1}},\dots,\overline{\sigma_{n}}\\}$,
where $\overline{\sigma_{k}}$ is as in Proposition 2.7. ∎

Following Kwak and Lee [40], a ring $R$ is called CN if $N(R[x])\subseteq
N(R)[x]$. Both NI rings and Armendariz rings are CN but not conversely. The
classes of quasi-Armendariz and CN rings are independent of each other
(Kheradmand et al. [33, Examples 3.15(1) and (2)]). Bhattacharjee [9] showed
that if a ring $R$ is $CN$ and quasi-Armendariz, then there is an equivalence
of the RNP property between the ring $R$ and its ring of polynomials with
coefficients in $R$.

Motivated by the previous facts, we introduce the following notion with the
aim of study the RNP property on Ore extensions and skew PBW extensions.

######  Definition 4.13.

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a skew PBW extension
over $R$. $R$ is said to be $\Sigma$-_skew CN_ if $N(A)\subseteq N(R)A$, where
$N(R)A:=\\{f=\sum_{i=0}^{t}a_{i}X_{i}\in A\mid a_{i}\in N(R)$, for $0\leq
i\leq t\\}$.

Hashemi et al. [23], studied the connections of the prime radical and the
upper nil radical of $R$ with the prime radical and the upper nil radical of
the extension skew PBW. For instance, some results establish sufficient
conditions to guarantee that $N^{*}(A)\subseteq N^{*}(R)A$, for a bijective
skew PBW extension $A$ over a $(\Sigma,\Delta)$-compatible ring $R$ [23,
Theorem 3.15]. In this way, if $A$ is NI, it follows that $R$ is $\Sigma$-skew
CN. Other important result states that if $A$ is a PBW extension over a
$\Sigma$-compatible ring $R$ and $N(R)$ is a $\Delta$-ideal, then
$N(A)\subseteq N(R)A$ [23, Proposition 4.1], that is, $R$ is a $\Sigma$-skew
CN. From [23, Corollary 4.12], it follows that if $R$ is a 2-primal
$\Sigma$-compatible ring, then $R$ is $\Sigma$-skew CN.

Reyes and Suárez [64] defined and studied the weak
$(\Sigma,\Delta)$-compatible property for skew PBW extensions. In particular,
it was proved that if $A$ is a skew PBW extension over a weak
$(\Sigma,\Delta)$-compatible NI ring $R$, then $f=\sum_{i=0}^{t}a_{i}X_{i}\in
N(A)$ if and only if $a_{i}\in N(R)$, for $0\leq i\leq t$ [64, Theorem 4.6].
Thus, if $f\in N(A)$, then $a_{i}\in N(R)$, for $0\leq i\leq t$, whence $f\in
N(R)A$. This means that if $A$ is a skew PBW extension over a weak
$(\Sigma,\Delta)$-compatible NI ring $R$, then $A$ is $\Sigma$-skew CN.

######  Example 4.14.

We present some examples of $\Sigma$-skew CN rings.

  * (1)

Let $q,h\in\Bbbk$, $q\neq 0$. Consider the algebra of q-differential operators
$D_{q,h}[x,y]=\Bbbk[y][x,\sigma,\delta]$ with $\sigma(y)=qy$ and
$\delta(y)=h$. This class of algebras satisfies that $N(D_{q,h}[x,y])\subseteq
N(\Bbbk[y])[x,\sigma,\delta]$. Since Ore extensions of injective type are skew
PBW extensions, we have other more general examples of Ore extensions that are
$\Sigma$-skew CN: if $R$ is a $\sigma$-compatible ring and $N(R)$ a
$\delta$-ideal of $R$, then by [57, Proposition 2.2],
$N(R[x;\sigma,\delta])\subseteq N(R)[x;\sigma,\delta]$, i.e. $R$ is
$\Sigma$-skew CN. If $R[x;\sigma,\delta]$ is NI and $N(R)$ is $\sigma$-rigid
then by [58, Theorem 3.1], we have that
$N(R[x;\sigma,\delta])=N(R)[x;\sigma,\delta]$, so $R$ is $\Sigma$-skew CN.

  * (2)

If $R$ is a commutative ring and $\mathfrak{g}$ is a finite dimensional Lie
algebra over $R$ with basis $\left\\{x_{1},\dots,x_{n}\right\\}$, then the
universal enveloping algebra of $\mathfrak{g}$, denoted $U(\mathfrak{g})$, is
a PBW extension over $R$. The set of nilpotent elements of the enveloping
algebra satisfies $N(A)=N(U(\mathfrak{g}))\cong N(\sigma(R)\left\langle
x_{1},\dots,x_{n}\right\rangle)\subseteq N(R)A$.

  * (3)

Consider the ring $A$ presented in Example 4.4. Let $f\in A$, such that
$f^{l}=0$, for some $l\in\mathbb{N}$, where
$f=a_{m}X_{m}+\cdots+a_{1}X_{1}+a_{0}=a_{m}X_{m}+h$ with
$deg(X_{i})=\alpha_{i}$ and $h=a_{m-1}X_{m}+\cdots+a_{0}$. Each $X_{i}$ is an
element of the set $\left\\{\prod_{i,j,k}X_{i,j}^{\alpha_{k}}\ |\ 1\leq k\leq
6\right\\}$. Thus, we have that
$0=f^{l}=(a_{m}X_{m}+\Delta)^{l}=a_{m}\prod_{k=1}^{l-1}\sigma_{i,j}^{k\alpha_{m}}(a_{m})X_{m}+\
{\rm lower\ terms}$. Since
$a_{m}\prod_{k=1}^{l-1}\sigma_{i,j}^{k\alpha_{m}}(a_{m})=0$, it follows that
$a_{m}^{l}=0$, by using $\Sigma$-compatibility. Hence, we obtain $a_{m}\in
N(R)=\left\langle z\right\rangle$. On the other hand, we can observe that
$0=f^{l}=(a_{m}X_{m}+h)^{l}=Q+h^{l}$, where $h=a_{m-1}X_{m}+\cdots+a_{0}$ and
each coefficient of $Q$ can be expressed in $a_{m}$ and
$\sigma_{i,j}^{k\beta}(a_{m})$, for some $\beta\in\mathbb{N}^{6}$ with
$k\in\mathbb{N}$. From the above, it follows that $h^{l}\in N(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$. Finally, if
$h={m-1}X_{m}+\cdots+a_{0}$, then
$h^{l}=(a_{m-1}X_{m}+\cdots+a_{0})^{l}=a_{m-1}\prod_{k=1}^{l-1}\sigma_{i,j}^{k\alpha_{m-1}}(a_{m-1})X_{m-1}+\
{\rm lower\ terms}$. By the same previous argument, we have $a_{m-1}\in N(R)$.
By continuing this process, we can see that $a_{i}\in N(R)=\left\langle
z\right\rangle$, for all $0\leq i\leq m$. Therefore, we have $f\in
N(R)\left\langle
x_{1,0},x_{1,1},x_{1,2},x_{2,0},x_{2,1},x_{2,2}\right\rangle$, that is, $R$ is
a $\Sigma$-skew CN ring.

######  Theorem 4.15.

Let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a skew PBW extension over
a $(\Sigma,\Delta)$-compatible and $\Sigma$-skew CN ring $R$ that satisfies
the (SQA1) condition. If the conditions established in Proposition 2.7 hold,
then $R$ is right $\Sigma$-skew RNP if and only if $A$ is right
$\overline{\Sigma}$-skew reflexive RNP.

###### Proof.

Let $R$ be a right $\Sigma$-skew RNP ring and consider two nilpotent elements
of $A$, that is, $f=\sum_{i=0}^{t}a_{i}X_{i}$, $g=\sum_{j=0}^{m}b_{j}Y_{j}\in
N(A)$ with ${\rm deg}(X_{i})=\alpha_{i}$, for $1\leq i\leq t$ and ${\rm
deg}(Y_{j})=\beta_{j}$, for $1\leq j\leq m$, respectively, such that $fAg=0$.
Since $R$ satisfies (SQA1) condition, then $a_{i}Rb_{j}=0$, for all $i,j$. On
the other hand, $R$ is $\Sigma$-skew CN, that is, $f,g\in N(R)A$, which
implies that $a_{i},b_{j}\in N(R)$ for all $i,j$. By using the $\Sigma$-skew
RNP property of $R$, we have $b_{j}R\sigma^{\alpha}(a_{i})=0$, for all $i,j$
and $\alpha\in\mathbb{N}^{n}\ \backslash\ \\{0\\}$. Consider an element $h\in
A$. We can note that each coefficient of $gh\overline{\sigma}^{\alpha}(f)$ are
products of the coefficient $b_{j}$ with elements of $R$ and several
evaluations of $a_{i}$ in $\sigma$’s and $\delta$’s depending of the
coordinates of $\alpha_{i},\beta_{j}$. From the previous statement and the
fact that $b_{j}R\sigma^{\alpha}(a_{i})=0$, by using of the
$(\Sigma,\Delta)$-compatibility of $R$, it follows that
$gh\overline{\sigma}^{\alpha}(f)=0$, for all $\alpha\in\mathbb{N}\ \backslash\
\\{0\\}$. Since $h$ is an arbitrary element of $A$, we have
$gA\overline{\sigma}^{\alpha}(f)=0$. Hence, $A$ is right
$\overline{\Sigma}$-skew RNP.

Conversely, suppose that $A$ is a right $\overline{\Sigma}$-skew RNP ring and
$a,b\in N(R)$ such that $aRb=0$. Since $R$ is $(\Sigma,\Delta)$-compatible, we
have $aAb=0$. This means that $bA\overline{\sigma}^{\alpha}(a)=0$, for all
$\alpha\in\mathbb{N}^{n}$ entailing $bR\sigma^{\alpha}(a)=0$, by using RNP
property of $A$ and $\overline{\sigma}^{\alpha}(a)\in N(R)$. Therefore, $R$ is
right $\Sigma$-skew RNP. ∎

######  Corollary 4.16.

([9, Corollary 2.13]) If $R$ is an Armendariz ring with an endomorphism
$\sigma$, then, $R$ is right $\sigma$-skew RNP if and only if $R[x]$ is right
$\overline{\sigma}$-skew RNP.

######  Corollary 4.17.

Let $R[x;\sigma]$ be a skew polynomial ring and $\sigma^{\prime}$ be an
endomorphism of $R$. If $R$ satisfies the (SQA1) condition, is skew CN and
$\sigma$-compatible, then $R$ is right $\sigma^{\prime}$-skew RNP if and only
if $R[x;\sigma]$ is right $\overline{\sigma^{\prime}}$-skew RNP.

######  Corollary 4.18.

([9, Proposition 2.12]) Let $R$ be a ring with an endomorphism $\sigma$.
Suppose that $R$ is quasi-Armendariz and CN. Then, $R$ is right $\sigma$-skew
RNP if and only if $R[x]$ is right $\overline{\sigma}$-skew RNP.

######  Proposition 4.19.

If $A$ is a skew PBW extension over a $(\Sigma,\Delta)$-compatible NI ring
$R$, then $A$ is nil-reflexive if and only if $A$ is RNP.

###### Proof.

Let $A$ be a skew PBW extension over $R$. Notice that $R$ is weak
$(\Sigma,\Delta)$-compatible. Thus, if $R$ is NI, from [68, Theorem 3.3] we
obtain that $A$ is NI, whence $N^{*}(A)=N(A)$. By [34, Proposition 2.1], we
have that $A$ is nil-reflexive if and only if $fAg=0$ implies $gAf=0$, for
elements $f,g\in N^{*}(A)$. Hence, $fAg=0$ implies $gAf=0$, for elements
$f,g\in N(A)$. Therefore, $A$ is nil-reflexive if and only if $A$ is RNP. ∎

######  Corollary 4.20.

Let $A$ be a skew PBW extension over an NI ring $R$ and suppose that the
conditions established in Proposition 2.7 hold. If $A$ is
$(\overline{\Sigma},\overline{\Delta})$-compatible, then $A$ is nil-reflexive
if and only if $A$ is $\overline{\Sigma}$-skew RNP.

###### Proof.

The assertion follows from Propositions 4.7 and 4.19. ∎

Next, we recall the following result which describes the Ore localization by
regular elements of the ring $R$ over a skew PBW extension.

######  Proposition 4.21.

([48, Lemma 2.6]) Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a skew
PBW extension over $R$, and let $S$ be the set of regular elements of $R$ such
that $\sigma_{i}(S)=S$, for every $1\leq i\leq n$, where $\sigma_{i}$ is
defined by Proposition 2.2.

  * (1)

If $S^{-1}R$ exists, then $S^{-1}A$ exists and it is a bijective skew PBW
extension over $S^{-1}R$ with $S^{-1}A=\sigma(S^{-1}R)\left\langle
x_{1}^{\prime},\dots,x_{n}^{\prime}\right\rangle$, where
$x_{i}^{\prime}:=\frac{x_{i}}{1}$ and the system of constants of $S^{-1}R$ is
given by $d_{i,j}^{\prime}:=\frac{d_{i,j}}{1}$,
$c_{i,\frac{r}{s}}^{\prime}:=\frac{\sigma_{i}(r)}{\sigma_{i}(s)}$, for all
$1\leq i,j\leq n$.

  * (2)

If $RS^{-1}$ exists, then $AS^{-1}$ exists and it is a bijective skew PBW
extension over $RS^{-1}$ with $AS^{-1}=\sigma(RS^{-1})\left\langle
x_{1}^{\prime},\dots,x_{n}^{\prime}\right\rangle$, where
$x_{i}^{\prime\prime}:=\frac{x_{i}}{1}$ and the system of constants of
$S^{-1}R$ is given by $d_{i,j}^{\prime\prime}:=\frac{d_{i,j}}{1}$,
$c_{i,\frac{r}{s}}^{\prime\prime}:=\frac{\sigma_{i}(r)}{\sigma_{i}(s)}$, for
all $1\leq i,j\leq n$.

Related with Proposition 4.21, if $S$ is a multiplicatively closed subset of a
ring $R$ consisting of central regular elements, and $\sigma$ is an
automorphism of $R$ such that $\sigma(S)\subseteq S$, then the mapping
$\overline{\sigma}:S^{-1}R\rightarrow S^{-1}R$ defined by
$\overline{\sigma}(u^{-1}a)=\sigma(u)^{-1}\sigma(a)$ for $u\in S$ and $a\in
R$, induces an endomorphism of $S^{-1}R$. If $\Sigma$ is a set of
automorphisms over $R$, we denote $\Sigma_{S}$ the set of automorphisms over
$S^{-1}R$, induced by $\Sigma$.

######  Theorem 4.22.

Let $R$ be a ring with a finite family of automorphism $\Sigma$ and $S$ a
multiplicatively closed subset of $R$ consisting of central regular elements
such that $\sigma^{\alpha}(S)\subseteq S$, for every
$\alpha\in\mathbb{N}^{n}$. Then, $R$ is right $\Sigma$-skew RNP if and only if
$S^{-1}R$ is right $\Sigma_{S}$-skew RNP.

###### Proof.

Suppose that $R$ is a right $\Sigma$-skew RNP and let
$s_{1}^{-1}a,s_{2}^{-1}b\in N(S^{-1}R)$ such that
$(s_{1}^{-1}a)S^{-1}R(s_{2}^{-1}b)=0$, that is,
$(s_{1}^{-1}a)s^{-1}r(s_{2}^{-1}b)=0$, for all $s^{-1}r\in S^{-1}R$. Notice
that $N(S^{-1}R)=S^{-1}N(R)$, which implies that $a,b\in N(R)$. In addition,
$(s_{1}ss_{2})^{-1}(arb)=(s_{1}^{-1}a)s^{-1}r(s_{2}^{-1}b)=0$. By definition
of $S^{-1}R$, there is $c\in S$ such that $c$ is a central element of $R$ and
$0=(arb)c=a(rc)b$. Since $R$ is right $\Sigma$-skew RNP, it follows that
$b(rc)\sigma^{\alpha}(a)=0$, for all nonzero $\alpha$ in $\mathbb{N}^{n}$.
Therefore, we have that
$(s_{2}^{-1}b)(s^{-1}r)\overline{\sigma}^{\alpha}(s_{1}^{-1}a)=(s_{2}^{-1}b)(s^{-1}r)\sigma^{\alpha}(s_{1})^{-1}\sigma^{\alpha}(a)=(s_{2}s\sigma^{\alpha}(s_{1}))^{-1}(br\sigma^{\alpha}(a))=0$,
since $b(r)\sigma^{\alpha}(a)c=0$, for each nonzero $\alpha$ in
$\mathbb{N}^{n}$. Therefore, $S^{-1}R$ is right $\Sigma_{S}$-skew RNP.
Reciprocally, let $a,b\in N(R)$ such that $aRb=0$, that is, $arb=0$, for all
$r\in R$. We have that $(1^{-1})(arb)=(1^{-1}a)(1^{-1}r)(1^{-1}b)=0$. If
$S^{-1}R$ is right $\Sigma_{S}$-skew RNP, we have
$0=(1^{-1}b)(1^{-1}r)\overline{\sigma}^{\alpha}(1^{-1}a)=(1^{-1}b)(1^{-1}r)\sigma^{\alpha}(1)^{-1}\sigma^{\alpha}(a)=(1^{-1})(br\sigma^{\alpha}(a))$.
Thus, $br\sigma^{\alpha}(a)=0$, that is, $R$ is right $\Sigma$-skew RNP. ∎

######  Theorem 4.23.

Let $A=\sigma(R)\langle x_{1},\dotsc,x_{n}\rangle$ be a bijective skew PBW
extension over a $(\Sigma,\Delta)$-compatible ring $R$. Suppose that the
conditions established in Proposition 2.7 hold, and let $S$ be a
multiplicatively closed subset of $R$ consisting of central regular elements
such that $\overline{\sigma}^{\alpha}(S)\subseteq S$ and $S^{-1}R$ is a
$\overline{\Sigma}_{S}$-skew CN which satisfies the (SQA1) condition. Then,
$A$ is right $\overline{\Sigma}$-skew RNP if and only if $S^{-1}A$ is right
$\overline{\Sigma}_{S}$-skew RNP.

###### Proof.

Suppose $A$ is right $\overline{\Sigma}$-skew RNP. Since $S^{-1}R$ is a
$\Sigma_{S}$-skew CN ring, then $R$ is $\Sigma$-skew CN. Similarly, the
condition (SQA1) also transfers from $S^{-1}R$ to $R$. Thus, we have that $R$
is right $\Sigma$-skew RNP, by using Theorem 4.15. By hypothesis, $S$ be a
multiplicatively closed subset of $R$ consisting of central regular elements,
which means that $S^{-1}R$ is right $\Sigma_{S}$-skew RNP, by Theorem 4.22. In
addition, by Proposition 4.21, we have that $S^{-1}A$ is a bijective skew PBW
extension over $S^{-1}R$, that is, $S^{-1}A=\sigma(S^{-1}R)\left\langle
x_{1}^{\prime},\dots,x_{n}^{\prime}\right\rangle$. Since $R$ is
$(\Sigma,\Delta)$-compatible, then $S^{-1}R$ is
$(\overline{\Sigma},\overline{\Delta})$-compatible [62, Theorem 4.20]. Notice
that $S^{-1}A$ is a bijective skew PBW extension over $S^{-1}R$, where
$S^{-1}R$ is a $(\overline{\Sigma},\overline{\Delta})$-compatible
$\overline{\Sigma}_{S}$-skew CN ring which satisfies the (SQA1) condition.
Hence, since $S^{-1}R$ is right $\Sigma$-skew RNP, then $S^{-1}A$ is a right
$\overline{\Sigma}_{S}$-skew RNP ring, by Theorem 4.15. Reciprocally, if
$S^{-1}A$ is right $\overline{\Sigma}_{S}$-skew RNP, where
$S^{-1}A=\sigma(S^{-1}R)\left\langle
x_{1}^{\prime},\dots,x_{n}^{\prime}\right\rangle$, then $S^{-1}R$ is right
$\overline{\Sigma}_{S}$-skew RNP, because $S^{-1}R$ is
$(\overline{\Sigma},\overline{\Delta})$-compatible
$\overline{\Sigma}_{S}$-skew CN and satisfies the condition (SQA1). The above
statement follows from Theorem 4.15. Since $S^{-1}R$ is
$\overline{\Sigma}_{S}$-skew RNP, then $R$ is $\Sigma$-skew RNP. Finally, $R$
is a $(\Sigma,\Delta)$-compatible ring and inherits the $\Sigma$-skew CN
property and the condition (SQA1) from the ring $S^{-1}R$, which implies that
$A$ is a right $\overline{\Sigma}$-skew RNP ring, by using Theorem 4.15. ∎

## Acknowledgments

Research is supported by Grant HERMES CODE 52464, Department of Mathematics,
Faculty of Science, Universidad Nacional de Colombia - Sede Bogotá.

## References

  * [1] D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26(7) (1998) 2265-2275.
  * [2] D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27(6) (1999) 2847-2852.
  * [3] S. Annin, Associated primes over Ore extension rings, _J. Algebra Appl._ 3(2) (2004) 193-205.
  * [4] E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18(4) (1974) 470-473.
  * [5] V. A. Artamonov, Derivations of skew PBW extensions, Commun. Math. Stat. 3(4) (2015) 449-457.
  * [6] M. Başer, C. Y. Hong and T. K. Kwak, On Extended Reversible Rings, Algebra Colloq. 16(1) (2009) 37-48.
  * [7] H. E. Bell, Near-rings in which each element is a power of itself, Bull. Aust. Math. Soc. 2(3) (1970) 363-368.
  * [8] A. Bell and K. Goodearl, Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions, Pacific J. Math. 131(1) (1998) 13-37.
  * [9] A. Bhattacharjee, Reflexive-nilpotents-property skewed by ring endomorphisms, _Arab. J. Math._ 9(1) (2020) 63-72.
  * [10] G. F. Birkenmeier, H. E. Heatherly and E. K. Lee, Completely prime ideals and associated radicals, S.K. Jain, S.T. Rizvi (Eds.), Ring Theory (Granville, OH, 1992), World Scientific, Singapore and River Edge (1993) 102-129.
  * [11] G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra 29(2) (2001) 639-660.
  * [12] K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, CRM Barcelona, Advanced Courses in Mathematics (Birkhaüser Basel, 2002).
  * [13] S. Chhawchharia and M. B. Rege, Armendariz rings, _Proc. Japan. Acad. Ser.A Math. Sci._ 73(1) (1997) 14-17.
  * [14] P. M. Cohn, Reversible Rings, Bull. Lond. Math. Soc. 31(6) (1999) 641-648.
  * [15] R. C. Courter, Rings all of whose factor rings are semi-prime, Canad. Math. Bull. 12(4) (1969) 417-426.
  * [16] W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez and H. Venegas, Skew PBW Extensions. Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications (Springer Cham, 2020).
  * [17] C. Gallego and O. Lezama, Gröbner bases for ideals of $\sigma$-PBW extensions, _Comm. Algebra_ 39(1) (2011) 50-75.
  * [18] J. Y. Gómez y H. Suárez, Double Ore extensions versus graded skew PBW extensions, _Comm. Algebra_ 48(1) (2020) 185-197.
  * [19] K. R. Goodearl and R. B. Jr. Warfield, An Introduction to Noncommutative Noetherian Rings (Cambridge University Press, 2004).
  * [20] J. M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ. 32(1) (1990) 73-76.
  * [21] M. Hamidizadeh, E. Hashemi and A. Reyes, A classification of ring elements in skew PBW extensions over compatible rings, Int. Electron. J. Algebra 28(1) (2020) 75-97.
  * [22] E. Hashemi, Kh. Khalilnezhad and A. Alhevaz, $(\Sigma,\Delta)$-Compatible skew PBW extension ring, Kyungpook Math. J. 57(3) (2017) 401-417.
  * [23] E. Hashemi, Kh. Khalilnezhad and A. Alhevaz, Extensions of rings over 2-primal rings, Matematiche 74(1) (2019) 141-162.
  * [24] E. Hashemi, Kh. Khalilnezhad and M. Ghadiri, Baer and quasi-Baer properties of skew PBW extensions, J. Algebr. Syst. 7(1) (2019) 1-24.
  * [25] E. Hashemi and A. Moussavi, Polinomial extensions of quasi-Baer rings, Acta Math. Hungar. 107(3) (2005) 207-224.
  * [26] S. Higuera and A. Reyes, A survey on the fusible property for skew PBW extensions, J. Algebr. Syst. To appear.
  * [27] Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168(1) (2002) 45-52.
  * [28] C. Y. Hong, N. K. Kim and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151(3) (2000) 215-226.
  * [29] C. Y. Hong, N. K. Kim and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31(1) (2003) 103-122.
  * [30] C. Y. Hong and T. K. Kwak, On minimal strongly prime ideals, Comm. Algebra 28(10) (2000) 4867-4878.
  * [31] C. Huh, Y. Lee and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30(2) (2002) 751-761.
  * [32] S. U. Hwang, Y. C. Jeon and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302(1) (2006) 186-199.
  * [33] M. Kheradmand, H. Khabazian, T. K. Kwak and Y. Lee, Reflexive property restricted to nilpotents, J. Algebra Appl. 16(3) (2017) 1750044.
  * [34] M. Kheradmand, H. K. Kim, T. K. Kwak and Y. Lee, Reflexive property on nil ideals, J. Algebra Appl. 17(1) (2017) 1750022.
  * [35] J. Y. Kim, Certain rings whose simple singular modules are GP-injective, Proc. Japan Acad. Ser. A Math. Sci. 81(7) (2005) 125-128.
  * [36] J. Y. Kim and J. U. Baik, On idempotent reflexive rings, Kyungpook Math. J. 46(4) (2006) 597-601.
  * [37] N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223(2) (2000) 477-488.
  * [38] N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure and Appl. Algebra 185(1-3) (2003) 207-223.
  * [39] J. Krempa, Some examples of reduced rings, Algebra Colloq. 3(4) (1996) 289-300.
  * [40] T. K. Kwak and Y. Lee, Rings over which coefficients of nilpotent polynomials are nilpotent, Int. J. Algebra Comput. 21(5) (2011) 745762.
  * [41] T. K. Kwak and Y. Lee, Reflexive property of rings, Comm. Algebra 40(4) (2012) 1576-1594.
  * [42] T. K. Kwak, Y. Lee and S. J. Yun, Reflexive property skewed by ring endomorphisms, Korean J. Math. 22(2) (2014) 217-234.
  * [43] J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14(3) (1971) 359-368.
  * [44] T. K. Lee and T. L. Wong, On Armendariz rings, Houston J. Math. 29(3) 583-593 (2003)
  * [45] O. Lezama, Computation of point modules of finitely semi-graded rings, Comm. Algebra 48(2) (2020) 862-878.
  * [46] O. Lezama, Some open problems in the context of skew PBW extensions and semi-graded rings, Commun. Math. Stat. 9(3) (2021) 347-378.
  * [47] O. Lezama, J. P. Acosta and A. Reyes, Prime ideals of skew PBW extensions, _Rev. Un. Mat. Argentina_ 56(2) (2015) 39-55.
  * [48] O. Lezama, J. P. Acosta, C. Chaparro, I. Ojeda and C. Venegas, Ore and Goldie theorems for skew PBW extensions, _Asian-Eur. J. Math._ 6(4) (2013) 1350061.
  * [49] O. Lezama and C. Gallego, $d$-Hermite rings and skew PBW extensions, _São Paulo J. Math. Sci._ 10(1) (2016) 60-72.
  * [50] O. Lezama and J. Gómez, Koszulity and point modules of finitely semi-graded rings and algebras, Symmetry 11(7) (2019) 1-22.
  * [51] O. Lezama and A. Reyes, Some homological properties of skew PBW extensions, Comm. Algebra 42(3) (2014) 1200-1230.
  * [52] G. Marks, On 2-primal Ore extensions, _Comm. Algebra_ 29(5) (2001) 2113-2123.
  * [53] G. Marks, Reversible and symmetric rings, J. Pure and Appl. Algebra 174(3) (2002) 311-318.
  * [54] G. Mason, Reflexive ideals, Comm. Algebra 9(17) (1981) 1709-1724.
  * [55] J. McConnell and J. Robson, Noncommutative Noetherian Rings. Graduate Studies in Mathematics, Second Edition (AMS, 2001).
  * [56] L. Motais de Narbonne. Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents. In: Proceedings of the 106th National Congress of Learned Societies, Perpignan, Paris: Bib. Nat., (1981) 71-73.
  * [57] A. R. Nasr-Isfahani, Ore extensions of 2-primal rings, _J. Algebra Appl._ 13(3) (2014) 1350117.
  * [58] A. R. Nasr-Isfahani, On NI skew polynomial ring, _Comm. Algebra_ 43(12) (2015) 5113-5120.
  * [59] O. Ore, Theory of non-commutative polynomials, Ann. of Math. (2) 34(3) (1933) 480-508.
  * [60] M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Jpn. Acad. Ser. A Math. Sci. 73(1) (1997) 14-17.
  * [61] A. Reyes and H. Suárez, $\sigma$-PBW extensions of skew Armendariz rings, _Adv. Appl. Clifford Algebr._ 27(4) (2017) 3197-3224.
  * [62] A. Reyes and H. Suárez, A notion of compatibility for Armendariz and Baer properties over skew PBW extensions, _Rev. Un. Mat. Argentina_ 59(1) (2018) 157-178.
  * [63] A. Reyes and H. Suárez, Skew Poincaré-Birkhoff-Witt extensions over weak zip rings, _Beitr. Algebra Geom._ 60(2) (2019) 197-216.
  * [64] A. Reyes and H. Suárez, Skew Poincaré-Birkhoff-Witt extensions over weak compatible rings, J. Algebra Appl. 19(12) (2020) 2050225.
  * [65] A. Reyes and H. Suárez, Radicals and Köthe’s conjecture for skew PBW extensions, Commun. Math. Stat. 9(2) (2021) 119-138.
  * [66] G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973) 43-60.
  * [67] H. Suárez, Koszulity for graded skew PBW extensions, _Comm. Algebra_ 45(10) (2017) 4569-4580.
  * [68] H. Suárez, A. Chacón and A. Reyes, On NI and NJ skew PBW extensions, arXiv:2106.09793v1[math.RA] (2021).
  * [69] A. B. Tumwesigye, J. Richter and S. Silvestrov, Centralizers in PBW Extensions. In: Silvestrov S., Malyarenko A., Rancic M. (eds) Algebraic Structures and Applications. SPAS 2017, Springer Proceedings in Mathematics & Statistics, Vol. 317, Springer (2020).
  * [70] C. Venegas, Automorphisms for skew PBW extensions and skew quantum polynomial rings, Comm. Algebra 43(5) (2015) 1977-1897.