Radicals and Köthe’s Conjecture for Skew PBW Extensions

Abstract

The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincaré–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Köthe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

1 Introduction

The noncommutative rings of interest for us in this article are the skew PBW (Poincaré–Birkhoff–Witt) extensions (also known as $\sigma$-PBW extensions) introduced in [6]. These algebraic structures were defined with the aim of generalizing the PBW extensions defined by Bell and Goodearl [5]. However, in the literature (c.f. [16]) it has been shown that skew PBW extensions also generalize several families of noncommutative rings of interest in representation theory, noncommutative algebraic geometry and mathematical physics, such as the following: iterated Ore extensions of injective type defined by Ore [21], almost normalizing extensions defined by McConnell and Robson [18], solvable polynomial rings introduced by Kandri-Rody and Weispfenning [10], diffusion algebras introduced by Isaev et al. [9] and studied by Hinchcliffe [8], the skew polynomial rings studied by Kirkman et al. [11], three-dimensional skew polynomial algebras [29], and more recently, the kind of Ore extensions investigated in [4]. The advantage of skew PBW extensions is that they do not require the coefficients to commute with the variables and, moreover, the coefficients need not come from a field. In fact, skew PBW extensions contain well-known groups of algebras such as some types Auslander–Gorenstein rings, some Calabi–Yau and skew Calabi–Yau algebras, some Artin–Schelter regular algebras, some Koszul algebras, quantum polynomials, some quantum universal enveloping algebras, some examples of G-algebras (see [31, 32, 37] for more details) and many other algebras of interest for modern mathematical physicists (see [16, 30] for a list of examples). For several relations between skew PBW extensions and another algebras with PBW bases, see [16, 29, 37].

Since ring and homological–theoretical properties of skew PBW extensions have been studied by the authors and others (cf. [3, 7, 23, 26, 31, 32, 34, 37]), and considering the work developed in [19] about radicals for Ore extensions, we think that an important task is to investigate these radicals in the context of skew PBW extensions and hence to obtain results about radicals for a class of noncommutative rings more general than (iterated) Ore extensions. With this in mind, the aim of this paper is to investigate several radicals of these extensions (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) under certain assumptions ($\varSigma$-skew Armendariz and $(\varSigma ,\varDelta )$-compatible, conditions introduced by the authors in [28, 33], respectively), and as a consequence, we prove that Köthe’s conjecture holds for skew PBW extensions under certain assumptions on R, which shows that this conjecture is true for a considerable number of noncommutative rings that cannot be expressed as iterated Ore extensions. As a matter of fact, we also establish some results about the question on the transfer of ring properties (reduced, symmetric, reversible and 2-primal) from a ring of coefficients R to a skew PBW extension A over R.

We present the structure of the article. In Sect. 2, we establish some useful results about skew PBW extensions for the rest of the paper. We recall the notions of $\varSigma$-skew Armendariz and $(\varSigma ,\varDelta )$-compatible rings (see Definitions 2.9 and 2.11, respectively) together with some key properties of these rings, and a list of remarkable rings which satisfy these conditions (Example 2.13). Next, in Sect. 3, we establish our more important results about the radicals above mentioned and the minimal prime ideals of skew PBW extensions under the assumptions of $\varSigma$-skew Armendariz and $(\varSigma ,\varDelta )$-compatibility. Then we prove that skew PBW extensions satisfy the Köthe’s conjecture (Propositions 3.17 and 3.19 ). Finally, in Sect. 4, we investigate the transfer of the properties of being 2-primal, Dedekind finite, IFP, reversible, symmetric and $ZC_l$, from a ring of coefficients R to a skew PBW extension A over R. As a matter of fact, the results presented in this paper are new for skew PBW extensions and all of them are similar to others existing in the literature for Ore extensions (actually, several proofs use similar arguments established in the literature, for instance [19]).

Throughout the paper, the word ring means an associative ring with unity not necessarily commutative.

2 Definitions and Elementary Properties

We start recalling the definition and some properties about our object of study.

Definition 2.1

([6], Definition 1) Let R and A be rings. We say that A is a skew PBW extension over R (also called $\sigma$-PBW extension of R), if the following conditions hold:

• (i) R is a subring of A sharing the same multiplicative identity element (R is said to be the coefficients ring);

• (ii) there exist elements $x_1,\ldots ,x_n\in A$ such that A is a left free R-module with basis ${\mathrm{Mon}}(A):= \{ x^{\alpha }=x_1^{\alpha _1}\cdots x_n^{\alpha _n}\mid \alpha =(\alpha _1,\ldots ,\alpha _n)\in {\mathbb {N}}^n\}$, and $x_1^{0}\cdots x_n^{0}:=1\in {\mathrm{Mon}}(A)$.

• (iii) For each $1\le i\le n$ and any $r\in R\ \backslash \ \{0\}$, there exists an element $c_{i,r}\in R\ \backslash \ \{0\}$ such that $x_ir-c_{i,r}x_i\in R$.

• (iv) For any elements $1\le i,j\le n$, there exists $c_{i,j}\in R\ \backslash \ \{0\}$ such that $x_jx_i-c_{i,j}x_ix_j\in R+Rx_1+\cdots +Rx_n$ (i.e., there exist elements $r_0^{(i,j)}, r_1^{(i,j)}$, $\cdots , r_n^{(i,j)} \in R$ with $x_jx_i - c_{i,j}x_ix_j = r_0^{(i,j)} + \sum _{k=1}^{n} r_k^{(i,j)}x_k$).

If all these conditions are satisfied, we will write $A:=\sigma (R)\langle x_1,\ldots ,x_n\rangle$. This notation is justified by the following proposition.

Proposition 2.2

([6], Proposition 3) Let A be a skew PBW extension over R. For each $1\le i\le n$, there exist an injective endomorphism $\sigma _i:R\rightarrow R$ and a $\sigma _i$-derivation $\delta _i:R\rightarrow R$, such that $x_ir=\sigma _i(r)x_i+\delta _i(r)$, for each $r\in R$. We will write $\varSigma :=\{\sigma _1,\cdots , \sigma _n\}$ and $\varDelta :=\{\delta _1,\cdots , \delta _n\}$.

Remark 2.3

With respect to the Definition 2.1 and the Proposition 2.2, we have the following facts:

• Since ${\mathrm{Mon}}(A)$ is a left R-basis of A, the elements $c_{i,r}$ and $c_{i,j}$ in Definition 2.1 are unique.

• In Definition 2.1 (iv), $c_{i,i}=1$. This follows from the equality $x_i^2-c_{i,i}x_i^2=s_0+s_1x_1+\cdots +s_nx_n$, with $s_j\in R$, which implies $1-c_{j,j}=0=s_j$.

From [6], Definition 4, a skew PBW extension A over a ring R will be called: (a) quasi-commutative, if the conditions (iii) and (iv) in Definition 2.1 are replaced by the following: (iii’) for each $1\le i\le n$ and all $r\in R\ \backslash \ \{0\}$, there exists $c_{i,r}\in R\ \backslash \ \{0\}$ such that $x_ir=c_{i,r}x_i$; (iv’) for any positive integers $1\le i,j\le n$, there exists $c_{i,j}\in R\ \backslash \ \{0\}$ such that $x_jx_i=c_{i,j}x_ix_j$. (b) bijective, if $\sigma _i$ is bijective for each $1\le i\le n$, and $c_{i,j}$ is invertible, for any $1\le i<j\le n$.

Definition 2.4

([6], Definition 6) Let A be a skew PBW extension over R with endomorphisms $\sigma _i$, $1\le i\le n$, as in Proposition 2.2.

• (i) For $\alpha =(\alpha _1,\ldots ,\alpha _n)\in {\mathbb {N}}^n$, $\sigma ^{\alpha }:=\sigma _1^{\alpha _1}\cdots \sigma _n^{\alpha _n}$, $|\alpha |:=\alpha _1+\cdots +\alpha _n$. If $\beta =(\beta _1,\dots ,\beta _n)\in {\mathbb {N}}^n$; then $\alpha +\beta :=(\alpha _1+\beta _1,\dots ,\alpha _n+\beta _n)$.

• (ii) For $X=x^{\alpha }\in {\mathrm{Mon}}(A)$, $\exp (X):=\alpha$, $\deg (X):=|\alpha |$, and $X_0:=1$. The symbol $\succeq$ will denote a total order defined on ${\mathrm{Mon}}(A)$ (a total order on ${\mathbb {N}}^n$). For an element $x^{\alpha }\in {\mathrm{Mon}}(A)$, ${\mathrm{exp}}(x^{\alpha }):=\alpha \in {\mathbb {N}}^n$. If $x^{\alpha }\succeq x^{\beta }$ but $x^{\alpha }\ne x^{\beta }$, we write $x^{\alpha }\succ x^{\beta }$. Every element $f\in A$ can be expressed uniquely as $f=a_0 + a_1X_1+\cdots +a_mX_m$, with $a_i\in R$, and $X_m\succ \cdots \succ X_1$. (Eventually, we will use the letters Y’s and $Z's$ to denote elements of ${\mathrm{Mon}}(A)$.) With this notation, we define ${\mathrm{lm}}(f):=X_m$, the leading monomial of f; ${\mathrm{lc}}(f):=a_m$, the leading coefficient of f; ${\mathrm{lt}}(f):=a_mX_m$, the leading term of f; ${\mathrm{exp}}(f):={\mathrm{exp}}(X_m)$, the order of f. Note that $\deg (f):={\mathrm{max}}\{\deg (X_i)\}_{i=1}^t$; $C_f:=\{a_0, a_1,\cdots , a_m\}$. If $f=0$, then ${\mathrm{lm}}(0):=0$, ${\mathrm{lc}}(0):=0$, ${\mathrm{lt}}(0):=0$. We also consider $X\succ 0$ for any $X\in {\mathrm{Mon}}(A)$. For a detailed description of monomial orders in skew PBW extensions, see [6], Sect. 3.

Proposition 2.5

([6], 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:

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

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

Proposition 2.6

([25], Proposition 2.9) If $\alpha =(\alpha _1,\cdots , \alpha _n)\in {\mathbb {N}}^{n}$ and r is an element of a ring R, then

$$\begin{aligned} x^{\alpha }r =&\ x_1^{\alpha _1}x_2^{\alpha _2}\cdots x_{n-1}^{\alpha _{n-1}}x_n^{\alpha _n}r = x_1^{\alpha _1}\cdots x_{n-1}^{\alpha _{n-1}}\biggl (\sum _{j=1}^{\alpha _n}x_n^{\alpha _{n}-j}\delta _n(\sigma _n^{j-1}(r))x_n^{j-1}\biggr )\\&+ \ x_1^{\alpha _1}\cdots x_{n-2}^{\alpha _{n-2}}\biggl (\sum _{j=1}^{\alpha _{n-1}}x_{n-1}^{\alpha _{n-1}-j}\delta _{n-1}(\sigma _{n-1}^{j-1}(\sigma _n^{\alpha _n}(r)))x_{n-1}^{j-1}\biggr )x_n^{\alpha _n}\\&+ \ x_1^{\alpha _1}\cdots x_{n-3}^{\alpha _{n-3}}\biggl (\sum _{j=1}^{\alpha _{n-2}} x_{n-2}^{\alpha _{n-2}-j}\delta _{n-2}(\sigma _{n-2}^{j-1}(\sigma _{n-1}^{\alpha _{n-1}}(\sigma _n^{\alpha _n}(r))))x_{n-2}^{j-1}\biggr )x_{n-1}^{\alpha _{n-1}}x_n^{\alpha _n}\\&+ \ \cdots + x_1^{\alpha _1}\biggl ( \sum _{j=1}^{\alpha _2}x_2^{\alpha _2-j}\delta _2(\sigma _2^{j-1}(\sigma _3^{\alpha _3}(\sigma _4^{\alpha _4}(\cdots (\sigma _n^{\alpha _n}(r))))))x_2^{j-1}\biggr )x_3^{\alpha _3}x_4^{\alpha _4}\cdots x_{n-1}^{\alpha _{n-1}}x_n^{\alpha _n} \\&+ \ \sigma _1^{\alpha _1}(\sigma _2^{\alpha _2}(\cdots (\sigma _n^{\alpha _n}(r))))x_1^{\alpha _1}\cdots x_n^{\alpha _n}, \ \ \ \ \ \ \ \ \ \ \sigma _j^{0}:={\mathrm{id}}_R\ \ {\mathrm{for}}\ \ 1\le j\le n. \end{aligned}$$

Remark 2.7

([25], Remark 2.10) If $a_i$ and $b_j$ are elements of R, for every i and j, and $X_i:=x_1^{\alpha _{i1}}\cdots x_n^{\alpha _{in}}$ and $Y_j:=x_1^{\beta _{j1}}\cdots x_n^{\beta _{jn}}$, when we compute every summand of $a_iX_ib_jY_j$ we obtain products of the coefficient $a_i$ with several evaluations of $b_j$ in $\sigma$’s and $\delta$’s depending of the coordinates of $\alpha _i$. This assertion follows from the expression:

$$\begin{aligned} a_iX_ib_jY_j =&\ a_i\sigma ^{\alpha _{i}}(b_j)x^{\alpha _i}x^{\beta _j} + a_ip_{\alpha _{i1}, \sigma _{i2}^{\alpha _{i2}}(\cdots (\sigma _{in}^{\alpha _{in}}(b_j)))} x_2^{\alpha _{i2}}\cdots x_n^{\alpha _{in}}x^{\beta _j} \\&+ \ a_i x_1^{\alpha _{i1}}p_{\alpha _{i2}, \sigma _3^{\alpha _{i3}}(\cdots (\sigma _{{in}}^{\alpha _{in}}(b_j)))} x_3^{\alpha _{i3}}\cdots x_n^{\alpha _{in}}x^{\beta _j} \\&+ \ a_i x_1^{\alpha _{i1}}x_2^{\alpha _{i2}}p_{\alpha _{i3}, \sigma _{i4}^{\alpha _{i4}} (\cdots (\sigma _{in}^{\alpha _{in}}(b_j)))} x_4^{\alpha _{i4}}\cdots x_n^{\alpha _{in}}x^{\beta _j}\\&+ \ \cdots + a_i x_1^{\alpha _{i1}}x_2^{\alpha _{i2}} \cdots x_{i(n-2)}^{\alpha _{i(n-2)}}p_{\alpha _{i(n-1)}, \sigma _{in}^{\alpha _{in}}(b_j)}x_n^{\alpha _{in}}x^{\beta _j} \\&+ \ a_i x_1^{\alpha _{i1}}\cdots x_{i(n-1)}^{\alpha _{i(n-1)}}p_{\alpha _{in}, b_j}x^{\beta _j}. \end{aligned}$$

Example 2.8

If $R[x_1;\sigma _1,\delta _1]\cdots [x_n;\sigma _n,\delta _n]$ is an iterated Ore extension where

• $\sigma _i$ is injective, for $1\le i\le n$;

• $\sigma _i(r)$, $\delta _i(r)\in R$, for every $r\in R$ and $1\le i\le n$;

• $\sigma _j(x_i)=cx_i+d$, for $i < j$, and $c, d\in R$, where c has a left inverse;

• $\delta _j(x_i)\in R + Rx_1 + \cdots + Rx_n$, for $i < j$,

then $R[x_1;\sigma _1,\delta _1]\cdots [x_n;\sigma _n, \delta _n] \cong \sigma (R)\langle x_1,\cdots , x_n\rangle$ ([16], p. 1212). Under these four conditions, skew PBW extensions are more general than Ore extensions of injective type (diffusion algebras, universal enveloping algebras of finite Lie algebras, and others, are examples of skew PBW extensions which cannot be expressed as iterated Ore extensions (see [16], Section 3.1, for more details). As we said in the Introduction, skew PBW extensions contains various well-known groups of algebras such as PBW extensions [5], the almost normalizing extensions [18], solvable polynomial rings [10], diffusion algebras [8, 9], three-dimensional skew polynomial algebras [29], some types of Auslander–Gorenstein rings, some skew Calabi–Yau algebras, some Artin–Schelter regular algebras, some Koszul algebras, quantum polynomials, some quantum universal enveloping algebras, the rings studied in [4], etc. Finally, note that quasi-commutative skew PBW extensions where $c_{i, r}=r$, for every $r\in R$ and $1\le i\le n$, coincide with the algebras studied by Kirkman et al. [11].

2.1 $\varSigma$-Skew Armendariz and $(\varSigma ,\varDelta )$-Compatible Rings

In commutative algebra, a ring B is called Armendariz (the term was introduced by Rege and Chhawchharia [22]), if whenever polynomials $f(x)=a_0+a_1x+\cdots + a_nx^n$, $g(x)=b_0+b_1x+\cdots + b_mx^m\in B[x]$ satisfy $f(x)g(x)=0$, then $a_ib_j=0$, for every i, j. The interest of this notion lies in its natural and its useful role in understanding the relation between the annihilators of the ring B and the annihilators of the polynomial ring B[x]. For instance, a reduced ring, i.e., a ring without nonzero nilpotent elements always satisfies this condition (actually, reduced rings are Abelian—that is, every idempotent is central-, and also semiprime, that is, its prime radical is trivial), see [25] for an adequate reference of this result. With respect to the noncommutative algebra, more exactly the well-known Ore extensions, the notion of Armendariz has been also studied. In this way, commutative and noncommutative treatments have been investigated in several papers, see [20, 28, 31] for a detailed list of references.

With the aim of formulating a skew notion of Armendariz ring for skew PBW extensions, in [28] the authors introduced the notion of $\varSigma$-skew Armendariz ring (Definition 2.9) with the aim of studying the properties Baer, quasi-Baer, p.p. and p.q.-Baer rings for these extensions. As a matter of fact, $\varSigma$-skew Armendariz rings generalize the class of $\varSigma$-rigid rings defined by the first author in [25] (see [28], Proposition 3.4). Let us say few words about the relation between these two classes of rings.

For a ring B with a ring endomorphism $\sigma :B\rightarrow B$, and a $\sigma$-derivation $\delta :B\rightarrow B$, Krempa [13] considered the Ore extension $B[x;\sigma ,\delta ]$ and defined $\sigma$ as a rigid endomorphism, if $b\sigma (b)=0$ implies $b=0$, for $b\in B$. Krempa called B a $\sigma$-rigid, if there exists a rigid endomorphism $\sigma$ of B. Properties of being Baer, quasi-Baer, p.p., and p.q.-Baer over $\sigma$-rigid rings have been investigated (c.f. [13] and others). All these results were generalized by the first author in [25] to the class of skew PBW extensions. There, the key fact was an adequate notion of rigidness for these extensions. Following [25], Definition 3.2, for a ring B and a family of endomorphisms $\varSigma$ of B, $\varSigma$ is called a rigid endomorphisms family, if $r\sigma ^{\alpha }(r)=0$ implies $r=0$, for every $r\in B$ and $\alpha \in {\mathbb {N}}^n$. B is called to be $\varSigma$-rigid, if there exists a rigid endomorphisms family $\varSigma$ of B. Note that if $\varSigma$ is a rigid endomorphisms family, then every element $\sigma _i\in \varSigma$ is a monomorphism. In this way, we can consider the family of injective endomorphisms $\varSigma$ and the family $\varDelta$ of $\varSigma$-derivations of a skew PBW extension A of a ring R (see Proposition 2.2). Note also that $\varSigma$-rigid rings are reduced rings: if B is a $\varSigma$-rigid ring and $r^2=0$, for $r\in B$, then $0=r\sigma ^{\alpha }(r^2)\sigma ^{\alpha }(\sigma ^{\alpha }(r))=r\sigma ^{\alpha }(r)\sigma ^{\alpha }(r)\sigma ^{\alpha }(\sigma ^{\alpha }(r))=r\sigma ^{\alpha }(r)\sigma ^{\alpha }(r\sigma ^{\alpha }(r))$, i.e., $r\sigma ^{\alpha }(r)=0$, and so $r=0$, that is, B is reduced. From [25], Proposition 3.5, we know that if A is a skew PBW extension of a ring R, then R is $\varSigma$-rigid if and only if A is a reduced ring. Other properties of $\varSigma$-rigid rings were established in [28, 31, 33, 34].

Next we recall the notion of $\varSigma$-skew Armendariz ring which is very important for the important results of the paper.

Definition 2.9

([28], Definitions 3.1 and 3.2) Let A be a skew PBW extension over a ring R. R is called a $\varSigma$-skew Armendariz ring, if for elements $f=\sum _{i=0}^{m} a_iX_i$ and $g=\sum _{j=1}^{t}b_jY_j$ of A, the equality $fg=0$ implies $a_i\sigma ^{\alpha _i}(b_j)=0$, for all $0\le i\le m$ and $0\le j\le t$. R is called a weak $\varSigma$-skew Armendariz ring, if for elements $f=\sum _{i=0}^{n} a_ix_i$ and $g=\sum _{j=1}^{n}b_jx_j$ of A, the equality $fg=0$ implies $a_i\sigma _i(b_j)=0$, for all $0\le i, j\le n$.

Note that every $\varSigma$-skew Armendariz ring is a weak $\varSigma$-skew Armendariz ring. Now, if A is a skew PBW extension over a $\varSigma$-rigid R, then R is $\varSigma$-skew Armendariz ([28], Proposition 3.4). The converse of this proposition is false as we can appreciate in [28], Remark 3.5. Nevertheless, we have the following equivalence of the notions of $\varSigma$-rigid and $\varSigma$-skew Armendariz assuming the reduceness of the ring.

Proposition 2.10

([28], Theorem 3.6) If A is a skew PBW extension over a ring R, then the following statements are equivalent: (1) R is reduced and $\varSigma$-skew Armendariz (2) R is $\varSigma$-rigid (3) A is reduced.

From [28], Proposition 3.9, we know that every weak $\varSigma$-skew Armendariz ring is Abelian, that is, all its idempotents are central.

Another generalization of $\varSigma$-rigid rings are the $(\varSigma ,\varDelta )$-compatible rings which were defined by the authors in [33]. In that paper, they extended the results presented in [25] about Baer, quasi-Baer, p.p. and p.q.-Baer to the class of compatible rings. Let us recall some key results of these rings.

Definition 2.11

([33], Definition 3.2) Consider a ring R with a finite family of endomorphisms $\varSigma$ and a finite family of $\varSigma$-derivations $\varDelta$. Then,

• 1. R is said to be $\varSigma$-compatible, if for each $a,b\in R$, $a((\sigma _1^{\alpha _1}\circ \cdots \circ \sigma _n^{\alpha _n})(b)) = a\sigma ^{\alpha }(b) = 0$ if and only if $ab=0$, for every $\alpha \in {\mathbb {N}}^{n}$;

• 2. R is said to be $\varDelta$-compatible, if for each $a,b \in R$, $ab=0$ implies that $a((\delta _1^{\beta _1} \circ \cdots \circ \delta _n^{\beta _n})(b)) =a\delta ^{\beta }(b)=0$, for every $\beta \in {\mathbb {N}}^{n}$.

If R is both $\varSigma$-compatible and $\varDelta$-compatible, R is called or $(\varSigma , \varDelta )$-compatible.

By [33], Proposition 3.4, if $\varSigma$ is a family of endomorphisms of a ring R, $\varDelta$ is a family of $\varSigma$-derivations of R, and R is a $\varSigma$-rigid ring, then R is $(\varSigma , \varDelta )$-compatible. The converse is false as [33], Example 3.6 shows. However, if the ring R is assumed to be reduced, these two notions coincide. More exactly,

Proposition 2.12

([33], Theorem 3.9) If A is a skew PBW extension over a ring R, then the following statements are equivalent: (i) R is reduced and $(\varSigma , \varDelta )$-compatible (ii) R is $\varSigma$-rigid (iii) A is reduced.

Example 2.13

From Propositions 2.10 and 2.12, we can observe that the concepts of $\varSigma$-rigid, $\varSigma$-skew Armendariz and $(\varSigma ,\varDelta )$-compatible coincide when R is a reduced ring. Since skew PBW extensions over domains are also domains ([16], Proposition 4.1), and every domain is a reduced ring, all these extensions are reduced rings and hence their coefficients ring is reduced, $\varSigma$-skew Armendariz, $(\varSigma ,\varDelta )$-compatible and $\varSigma$-rigid. Remarkable examples of skew PBW extensions over domains are the following (see [16, 30] for the definition and reference of every example and its characterization as a skew PBW extension): PBW extensions defined by Bell and Goodearl (which include the classical commutative polynomial rings, universal enveloping algebra of a Lie algebra, and others); the quantum plane ${\mathcal {O}}_q(\Bbbk ^{2})$; the algebra of q-differential operators $D_{q,h}[x,y]$; the mixed algebra $D_h$; the operator differential rings; the algebra of differential operators $D_\mathbf{q}(S_\mathbf{q})$ on a quantum space ${S_\mathbf{q}}$, some operator algebras (for example, the algebra of linear partial differential operators, the algebra of linear partial shift operators, the algebra of linear partial difference operators, the algebra of linear partial q-dilation operators, and the algebra of linear partial q-differential operators); the family of diffusion algebras; the quantum algebra ${\mathcal {U}}'(\mathfrak {so}(3,\Bbbk ))$; the family of three-dimensional skew polynomial algebras (there are exactly fifteen of these algebras, see [29]); Dispin algebra ${\mathcal {U}}(\mathrm{osp}(1,2))$; Woronowicz algebra ${\mathcal {W}}_v(\mathfrak {sl}(2,\Bbbk ))$; the complex algebra $V_q(\mathfrak {sl}_3({\mathbb {C}}))$; q-Heisenberg algebra $\mathbf{H}_n(q)$; the Hayashi algebra $W_q(J)$, and more. Now, as we said at the Introduction, several algebras of mathematical physics can be expressed as skew PBW extensions: Weyl algebras, additive and multiplicative analogue of the Weyl algebra, quantum Weyl algebras, q-Heisenberg algebra, and others, which allows us to characterize several properties with physical meaning. In fact, two noncommutative algebras studied in physics which are important examples of skew PBW extensions can be found in [35]. As a matter of fact, the notion of compatibility has been defined for modules over these extensions, see [27].

Next we present one result about $(\varSigma ,\varDelta )$-compatible rings. This result will be important in several proofs in the next section.

Proposition 2.14

([33], Proposition 3.8) Let R be a $(\varSigma , \varDelta )$-compatible ring. For every $a, b \in R$, we have:

• (1) if $ab=0$, then $a\sigma ^{\theta }(b) = \sigma ^{\theta }(a)b=0$, for each $\theta \in {\mathbb {N}}^{n}$.

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

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

Remark 2.15

In [28], Sect. 4 and [33], Sect. 4, the authors characterized weak $\varSigma$-Armendariz and $(\varSigma ,\varDelta )$-compatible rings for the classical ring of quotients of a skew PBW extension, so we can apply the results obtained in Sects. 3 and 4 for several algebras obtained as quotients of these extensions.

3 Radicals of Skew PBW Extensions

In this section, we investigate the radicals mentioned in the Introduction for skew PBW extensions. The results obtained in this section extend [19] and hence establish a general theory about radicals for noncommutative rings more general than (iterated) Ore extensions. Hence, we contribute to the study of ideals of these extensions. (This task has been developed partially in [15, 35].) We start recalling the condition (SA1) introduced in [33]:

Definition 3.1

([33], Definition 4.1) Let A be a skew PBW extension of R. We say that R satisfies the condition (SA1), if whenever $fg=0$ for $f=a_0 + a_1X_1 + \cdots + a_mX_m$ and $g=b_0 + b_1Y_1 + \cdots + b_tY_t$ elements of A, then $a_ib_j = 0$, for every i, j.

Every $\varSigma$-rigid ring satisfies the condition (SA1) ([25], Proposition 3.6). Proposition 3.2 establishes the relation between the notions $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, and the condition (SA1).

Proposition 3.2

If A is a skew PBW extension of a ring R,  then R is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz if and only if R satisfies (SA1).

Proof

Suppose that R is $(\varSigma , \varDelta )$-compatible and $(\varSigma ,\varDelta )$-skew Armendariz. Consider two elements $f = \sum _{i=0}^{m} a_iX_i$ and $g=\sum _{j=0}^{t} b_jY_j$ in A with $fg=0$. By assumption, $a_i\sigma ^{\alpha _i}(b_j)=0$, for every i, j. Using the $\varSigma$-compatibility of R, we obtain $a_ib_j=0$, for each i, j. Now, if $f, g \in A$ with $a_ib_j=0$, for $0\le i\le m$ and $0\le j\le t$, from Propositions 2.6 and 2.14 together with Remark 2.7, it follows that $fg=0$.

Conversely, assume that for the elements f, g of A given by $f = \sum _{i=0}^{m} a_iX_i$ and $g=\sum _{j=0}^{t} b_jY_j$, respectively, $fg=0$ if and only if $a_ib_j=0$, for every i, j. Let us see that R is $(\varSigma ,\varDelta )$-compatible. Let $a, b\in R$ with $ab=0$. By assumption on R, if $f=ax^{\alpha }$ (for any $\alpha \in {\mathbb {N}}$) and $g=b$, then $fg=0$, that is, $0=ax^{\alpha }(b) = a\sigma ^{\alpha }(b)x^{\alpha } + ap_{\alpha , b}$, whence $a\sigma ^{\alpha }(b) = a\delta ^{\alpha }(b)=0$. Now, if $a\sigma ^{\theta }(b)=0$, for any $\theta \in {\mathbb {N}}$, then $x^{\theta }a\sigma ^{\theta }(b)=0$, i.e., $(\sigma ^{\theta }(a)x^{\theta } + p_{\theta ,a})\sigma ^{\theta }(b)=0$, and so $\sigma ^{\theta }(ab)=0$ which shows that $ab=0$, since $\sigma ^{\theta }$ is injective. The proof that R is $\varSigma$-skew Armendariz is similar. $\square$

Proposition 3.3

If A is a skew PBW extension of a ring R, then R is $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz if and only if R satisfies (SA1).

Proof

Let R be $(\varSigma , \varDelta )$-compatible and weak $(\varSigma ,\varDelta )$-skew Armendariz. Consider two elements $f = \sum _{i=0}^{n} a_ix_i$ and $g=\sum _{j=0}^{n} b_jx_j$ in A with $fg=0$. By assumption, $a_i\sigma _{i}(b_j)=0$, for every i, j. Using the $\varSigma$-compatibility of R, we obtain $a_ib_j=0$, for each i, j. If $f, g \in A$ with $a_ib_j=0$, for $0\le i, j\le n$, Propositions 2.6 and 2.14 together with Remark 2.7, it follows that $fg=0$.

Assume that for the elements f, g of A given by $f = \sum _{i=0}^{n} a_ix_i$ and $g=\sum _{j=0}^{n} b_jx_j$, respectively, $fg=0$ if and only if $a_ib_j=0$, for every i, j. Let $a, b\in R$ with $ab=0$. By assumption on R, if $f=ax_i$, for any i, and $g=b$, then $fg=0$, that is, $0=ax_ib = a\sigma _i(b)x_i + a\delta _i(b)$, whence $a\sigma _i(b) = a\delta _i(b)=0$. Now, if $a\sigma _j(b)=0$, for any $1\le j\le n$, then $x_ja\sigma _j(b)=0$, i.e., $(\sigma _j(a)x_j + \delta _j(a))\sigma _j(b)=0$, and so $\sigma _j(ab)=0$ which shows that $ab=0$, since $\sigma _j$ is injective. The proof that R is weak $\varSigma$-skew Armendariz uses a similar argument. $\square$

Lemma 3.4

Let A be a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R. If $p_1,\cdots , p_l$ are elements of A such that $p_1\cdots p_l=0$, then $a_1\cdots a_l = 0$, where $a_i\in C_{f_i}$, for each i.

Proof

We proceed by induction following the notation considered in Proposition 2.5. If $l=2$, let $p_1=\sum _{i=0}^{m} a_iX_i,\ p_2=\sum _{j=0}^{t} b_jY_j$. By assumption we have $a_i\sigma ^{\alpha _i}(b_j)=0$, for every i, j, and using the $\varSigma$-compatibility of R, $a_ib_j = 0$, for every value of i and j, so the assertion follows.

Let $l > 2$. If $h:=p_2p_3\cdots p_l$, then $p_1h=0$, and by the reasoning above, $a_1a_h=0$, where $a_1\in C_{p_1},\ a_h\in C_{h}$. Having in mind the form of the elements of h, that is, $a_h =a_2\cdots a_l$, where $a_2\in C_{f_2}, \cdots , a_l\in C_{f_l}$ (which is due to the fact that R is $\varSigma$-skew Armendariz and $\varSigma$-compatible), then we obtain $a_1\cdots a_l=0$. $\square$

Next, we present different preliminary results following [19] with the aim of establishing analogous results for skew PBW extensions.

Lemma 3.5

Let A be a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz ring R. If $ab=c^{m}=0$, for some $m\in {\mathbb {N}}$ and every elements a, b, c of R, then $ac^{m-1}b=0$.

Proof

If $m=1$ the result is immediate. Take $m\ge 2$ and suppose that $ab=c^{m-1}=0$. Consider the elements $f=a-ac^{m-1}x_i$ and $g=b + c^{m-1}\delta _i(b) + c^{m-1}\sigma _i(b)x_i\in A$, for any $i\in \{1,\cdots , n\}$. We have

$$\begin{aligned} fg =&\ (a-ac^{m-1}x_i)(b + c^{m-1}\delta _i(b) + c^{m-1}\sigma _i(b)x_i)\\ =&\ ab + ac^{m-1}\delta _i(b) + ac^{m-1}\sigma _i(b)x_i \\&- \ ac^{m-1}x_ib - ac^{m-1}x_ic^{m-1}\delta _i(b) - ac^{m-1}x_ic^{m-1}\sigma _i(b)x_i\\ =&\ ab + ac^{m-1}\delta _i(b) + ac^{m-1}\sigma _i(b)x_i - ac^{m-1}\sigma _i(b)x_i - ac^{m-1}\delta _i(b)\\&- \ ac^{m-1}\sigma _i(c^{m-1}\delta _i(b)) - ac^{m-1}\delta _i(c^{m-1}\delta _i(b))\\&- \ ac^{m-1}\sigma _i(c^{m-1}\sigma _i(b))x_i^{2} - ac^{m-1}\delta _i(c^{m-1}\sigma _i(b))x_i\\ =&\ -ac^{m-1}\sigma _i(c^{m-1}\delta _i(b)) - ac^{m-1}\delta _i(c^{m-1}\delta _i(b))\\&- \ ac^{m-1}\sigma _i(c^{m-1}\delta _i(b))x_i^{2} - ac^{m-1}\delta _i(c^{m-1}\delta _i(b)). \end{aligned}$$

By assumption, $ab=c^{m-1}=0$, so $ac^{m-1}c^{m-1}b=0$, whence Proposition 2.14 implies that every one of the above terms is equal to zero, so $fg=0$. Since R is $\varSigma$-skew Armendariz, then $a(b+c^{m-1}\delta _i(b)) = ac^{m-1}\sigma _i(b)=0$ whence $ac^{m-1}x_ib= ac^{m-1}(\sigma _i(b)x_i+\delta _i(b))=0$. Therefore, Proposition 3.3 guarantees that $ac^{m-1}b=0$.

$\square$

Proposition 3.6

If R is a $(\varSigma ,\varDelta )$-compatible weak $\varSigma$-skew Armendariz ring, and $ab=c^{m} = 0$, for some positive integer m and every elements $a, b, c\in R$, then $acb=0$.

Proof

We use a similar argument to the used in Theorem 2.6, [19]. Note that if $m\ne 2^{k}$, then there exists a positive integer k with $2^{k} > m$ and $c^{2^{k}} = 0$. This fact means that it is enough to assume that $m=2^{k}$ and $ab=c^{m}=0$. From Lemma 3.5, we can see that $ab=(c^{2^{k-1}})^{2}=0$ implies $ac^{2^{k-1}}b=0$. With this in mind, consider the elements $f=a-ac^{2^{k-2}}x_i$ and $g=b+c^{2^{k-2}}\delta _i(b) + c^{2^{k-2}}\sigma _i(b)x_i$, for any $i\in \{1,\cdots , n\}$. Note that

$$\begin{aligned} fg =&\ (a-ac^{2^{k-2}}x_i)(b+c^{2^{k-2}}\delta _i(b) + c^{2^{k-2}}\sigma _i(b)x_i)\\ =&\ ab+ac^{2^{k-2}}\delta _i(b) + ac^{2^{k-2}}\sigma _i(b)x_i - ac^{2^{k-2}}x_ib \\&- \ ac^{2^{k-2}}x_ic^{2^{k-2}}\delta _i(b) - ac^{2^{k-2}}x_ic^{2^{k-2}}\sigma _i(b)x_i\\ =&\ ac^{2^{k-2}}\delta _i(b) + ac^{2^{k-2}}\sigma _i(b)x_i - ac^{2^{k-2}}\sigma _i(b)x_i - ac^{2^{k-2}}\delta _i(b)\\&- \ ac^{2^{k-2}}\sigma _i(c^{2^{k-2}}\delta _i(b))x_i - ac^{2^{k-2}}\delta _i(c^{2^{k-2}}\delta _i(b))\\&- \ ac^{2^{k-2}}\sigma _i(c^{2^{k-2}}\sigma _i(b))x_i^{2} - ac^{2^{k-2}}\delta _i(c^{2^{k-2}}\sigma _i(b))\delta _i(b)x_i. \end{aligned}$$

Having in mind that $ac^{2^{k-1}}b=0$, from Proposition 2.14 we conclude that $fg=0$. Now, from the assumptions on R and Proposition 3.3 we obtain that $a(b+c^{2^{k-2}}\delta _i(b)) = ac^{2^{k-2}}\sigma _i(b)=0$, and hence, $ac^{2^{k-2}}\sigma _i(b) = ac^{2^{k-2}}\delta _i(b)=0$, that is, $ac^{2^{k-2}}x_ib=0$ whence $ac^{2^{k-2}}b=0$.

If we repeat the above argument, we can prove that $(a-ac^{2^{k-3}}x_i)(b+c^{2^{k-3}}\delta _i(b) + c^{2^{k-3}}\sigma _i(b)x_i)=0$, which means $ac^{2^{k-3}}\sigma _i(b) = a(b+c^{2^{k-3}}\delta _i(b))=0$ and then $ac^{2^{k-3}}b=0$. Considering this reasoning $k-1$ times, we obtain $ac^{2^{k-(k-1)}}b=0$, and so $ac^{2}b=0$. Finally, for the polynomials $a-acx$ and $b+c\delta _i(b) + c\sigma _i(b)x_i$, we have $(a-acx_i)(b+c\delta _i(b) + c\sigma _i(b)x_i) = ab + ac\delta _i(b) + ac\sigma _i(b)x_i - acx_ib - acx_ic\delta _i(b) - acx_ic\sigma _i(b)x_i = ac\delta _i(b) + ac\sigma _i(b)x_i - ac\sigma _i(b)x_i - ac\delta _i(b) - ac\sigma _i(c\delta _i(b))x_i - ac\delta _i(c\delta _i(b)) - ac\sigma _i(c\sigma _i(b))x_i^{2} - ac\delta _i(c\sigma _i(b))x_i^{2}$. Since $ac^{2}b=0$, Proposition 2.14 shows that all these coefficients are equal to zero, so $(a-acx_i)(b+c\delta _i(b) + c\sigma _i(b)x_i) = 0$, whence by Proposition 3.3 we obtain $ac\sigma _i(b) = ac\delta _i(b) = 0$, and so $acx_ib=0$, or equivalently, $acb=0$. $\square$

Proposition 3.7

Let A be a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R. If for any elements $f, g, h\in A$, $f^{s}=0$, for some $s\in {\mathbb {N}}$, and $gh=0$, then $gfh=0$.

Proof

Consider the expressions for f, g and h given by $f=\sum _{i=0}^{m} a_iX_i, g=\sum _{j=0}^{p} b_jY_j$ and $h=\sum _{l=0}^{t}c_lZ_l$. Since $f^{s}=0$, from Proposition 3.4 we know that $a_i^{s}=0$, for every $0\le i\le m$. Now, by assumption $gh=0$, and using that R is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, we have $b_jc_l=0$, for every j, l. In this way, by Proposition 3.6 we can assert $b_ja_ic_l=0$, for each value of i, j, l. Finally, Propositions 2.6 and 2.14 with Remark 2.7 allow us to conclude that $gfh=0$. $\square$

Following the notation presented in [19], the Wedderburn radical (the largest nilpotent ideal in B), the lower nil radical (the intersection of all the prime ideals in R), the Levitzky radical (the sum of all locally nilpotent ideals), the upper nil radical (the sum of all nil ideals), the set of all nilpotent elements of B, and the sum of all nil left ideals of B (this set is equal to the sum of all nil right ideals of B) will be denoted by $N_0(B)$, $Nil_{*}(B)$, $L-rad(B)$, $Nil^{*}(B)$, Nil(B) and A(B), respectively.

The next theorem is the first important result of the paper.

Theorem 3.8

If A is a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then

$$\begin{aligned} N_0(A) = Nil_{*}(A) = L-rad(A) = Nil^{*}(A). \end{aligned}$$

Proof

It is sufficient to show that $Nil^{*}(A)\subseteq N_0(A)$. Let f be an element of $Nil^{*}(A)$. By definition, there exists $m\in {\mathbb {N}}$ such that $f^{m}=0$. Let us see that $(AfA)^{2m-1} = 0$. With this objective, note that $AfA\subseteq Nil^{*}(A)$ (since $f^{m-1}f=0$), and so for every $g\in AfA$, we obtain $f^{m-1}gf=0$ (Proposition 3.7). This fact means that $f^{m-1}AfAf=0$, whence $f^{m-2}fAfAf=0$. As we saw before, $f^{m-2}AfAfAfAf=0$. Repeating this argument we obtain $fAfAfAfA\cdots fAf$, and hence, $(AfA)^{2m-1}$ which concludes the proof. $\square$

The next theorem is the second important result of the paper.

Theorem 3.9

If R is a $(\varSigma , \varDelta )$-compatible and weak $\varSigma$-skew Armendariz ring, then $N_0(R) = Nil_{*}(R) = L-rad(R) = Nil^{*}(R)$.

Proof

The assertions follow from Proposition 3.6 using a similar reasoning to the established in Theorem 3.8. $\square$

With the aim of establishing another results above radicals of skew PBW extensions, we consider the following definition.

Definition 3.10

([15], Definition 2.1) Let A be a skew PBW extension over a ring R. Consider the sets of endomorphisms $\varSigma$ and $\varDelta$ in Proposition 2.2. (i) An ideal I of R is called $\varSigma$-invariant, if $\sigma _i(I)\subseteq I$, for every $1\le i\le n$. $\varDelta$-invariant ideals are defined similarly. If I is both $\varSigma$ and $\varDelta$-invariant, we say that I is $(\varSigma ,\varDelta )$-invariant. (ii) A proper $\varSigma$-invariant ideal of R is $\varSigma$-prime, if whenever a product of two $\varSigma$-invariant ideals is contained in I, one of the ideals is contained in I. $\varDelta$-prime and $(\varSigma ,\varDelta )$-prime ideals are defined similarly.

We write $P_{(\varSigma ,\varDelta )} := {\mathrm{Spec}}(R;\varSigma ,\varDelta )$ for the set of all $(\varSigma ,\varDelta )$-prime ideals of R and $Nil_{*}(R;\varSigma ,\varDelta ) = \bigcap _{P\in P_{(\varSigma ,\varDelta )}} P$ for the $(\varSigma ,\varDelta )$-prime radical of R. R is a $\varSigma$-prime (resp. $\varSigma$-semiprime) ring, if the ideal 0 is $\varSigma$-prime (resp. if $Nil_{*}(R;\varSigma ) = 0$). In a similar way, we define $\varDelta$-prime, $\varDelta$-semiprime, $(\varSigma ,\varDelta )$-prime and $(\varSigma ,\varDelta )$-semiprime rings. About these sets we have the following theorem.

Theorem 3.11

If R is a $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz ring, then $Nil_{*}(R;\varSigma ,\varDelta ) = Nil_{*}(R) = Nil_{*}(R;\varSigma ) = Nil_{*}(R;\varDelta )$.

Proof

From the definitions above, the inclusions $Nil_{*}(R;\varSigma ,\varDelta )\subseteq Nil_{*}(R;\varDelta )$ and $Nil_{*}(R;\varDelta )\subseteq Nil_{*}(R)$ follow, so $Nil_{*}(R;\varSigma ,\varDelta )\subseteq Nil_{*}(R)$. Consider an element a of $Nil_{*}(R)$. Theorem 3.9 says us that $a\in N_0(R)$ which means that $\langle a\rangle ^{m} = 0$, for some $m\in {\mathbb {N}}$, where $\langle a\rangle$ is the two-sided ideal of R generated by a. Let I be the $(\varSigma ,\varDelta )$-ideal generated by a. The equality $\langle a\rangle ^{m} = 0$ implies that $r_1ar_2a\cdots ar_mar_{m+1} = 0$, for any elements $r_1,\cdots , r_{m+1}\in R$ whence $I^m=0$ (Proposition 2.14). If $P\in \mathrm{Spec}(R;\varSigma ,\varDelta )$, it is clear that $I\subseteq P$ and so $a\in Nil_{*}(R;\varSigma ,\varDelta )$ which shows that $Nil_{*}(R;\varSigma ,\varDelta ) = Nil_{*}(R)$. Using a similar reasoning we can prove the equalities $Nil_{*}(R;\varSigma ) = Nil_{*}(R)$ and $Nil_{*}(R;\varDelta ) = Nil_{*}(R)$ which concludes the proof. $\square$

If $D\subseteq R$ and A is a skew PBW extension over R, then DA will denote the set of elements f of A with coefficients in D, that is, $DA:=\{f\in A\mid f= d_0 + d_1X_1+\cdots + d_mX_m,\ d_i\in D,\ {\mathrm{for every}}\ i\}$.

Theorem 3.12

If A is a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then

$$\begin{aligned} Nil_{*}(A) = Nil_{*}(R)A = Nil_{*}(R;\varSigma ,\varDelta )A = Nil_{*}(R;\varSigma )A = Nil_{*}(R;\varDelta )A. \end{aligned}$$

Proof

One can see that $Nil_{*}(R;\varSigma ,\varDelta )A\subseteq Nil_{*}(A)$, and together with Theorem 3.11, we can assert that $Nil_{*}(R)A = Nil_{*}(R;\varSigma ,\varDelta )A\subseteq Nil_{*}(A)$. On the other hand, let $f\in Nil_{*}(A)$ given by $f=\sum _{i=0}^{m} a_iX_i$. From Theorem 3.8, we obtain that $f\in Nil_0(A)$, that is $\langle f\rangle ^{t}=0$, for some $t\in {\mathbb {N}}$, where $\langle f\rangle$ is the two-sided ideal of A generated by f. In this way, Lemma 3.4 guarantees that $(Ra_iR)^{t} = 0$, for every value of i, which means that $a_i\in Nil_{*}(R)$. Hence $f\in Nil_{*}(R)A$, and so $Nil_{*}(A) = Nil_{*}(R)A$. The other equalities follow from Theorem 3.11. $\square$

The above results guarantee the following fact.

Corollary 3.13

If A is a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then we have the following equalities:

$$\begin{aligned} N_0(R)A =&\ N_0(A) = Nil_{*}(A) = Nil_{*}(R)A\\ =&\ L-rad (A) = L-rad(R)A = Nil^{*}(A)\\ =&\ Nil_{*}(R)A = Nil_{*}(R;\varSigma ,\varDelta )A = Nil_{*}(R;\varSigma )A = Nil_{*}(R;\varDelta )A. \end{aligned}$$

Remark 3.14

If R is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then the following statements are equivalent: R is semiprime; R is $(\varSigma ;\varDelta )$-semiprime; R is $\varSigma$-semiprime; R is $\varDelta$-semiprime; A is semiprime.

Amitsur [1] asked the following question: if B is a nil ring, is the polynomials ring B[x]? A negative answer to this question was given by Smoktunowicz [36]. Nevertheless, this question formulated for skew PBW extensions over $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz rings has a positive answer as the following result shows.

Proposition 3.15

If A is a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then A is a nil ring.

Proof

The assertion follows from results above. $\square$

Proposition 3.16

If R is a semiprime $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz, then R has no nonzero nil one-sided ideal.

Proof

We follow the ideas presented in [19], Proposition 2.16. Suppose that R has a nonzero nil one-sided ideal. Let I be a nonzero nil right ideal of R (the left case is similar). In the case that there exists a nonzero element a of I with $a^{m}=0$ and $a^{m-1}\ne 0$, for some $m>2$, then every element in aR is nilpotent, since aR is a nil right ideal. Hence, $aaRa^{m-1}=0$ (Proposition 3.6) which is obviously a contradiction. Then, for every $r\in I$, $r^{2}=0$, and so there exists $0\ne a\in I$ with $a^{2}=0$. Let us see that $(aR)^{3}=0$. If $r_1, r_2, r_3$ are elements of R, then $ar_2r_1ar_2r_1=0$ ($ar_2r_1\in I$). So, $(r_1ar_2)^{3}=0$ and $a^{2}=0$. Theorem 3.6 guarantees that $ar_1ar_2a=0$ which implies that $ar_1ar_2ar_3=0$, that is, $(aR)^{3}=0$, and using that R is semiprime, $aR=0$ whence $a=0$, a contradiction. Therefore our initial assumption is false and R has no nonzero nil right ideal. $\square$

The Köthe’s conjecture posed in 1930 conjectured that a ring B with no nonzero nil (two-sided) ideals would also have no nonzero nil one-sided ideals (see [12] for the original formulation and [14] for equivalent statements of the conjecture). It is known that this conjecture holds for several classes of rings such as Noetherian rings (both left and right Noetherian), Goldie rings, rings with right Krull dimension, monomial algebras, PI rings and algebras over uncountable fields. Propositions 3.17 and 3.19 establish that the conjecture is also true for coefficients ring which are $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz, and skew PBW extensions over these rings. As a matter of fact, in [24] it was investigated the Jacobson’s conjecture for these extensions.

Proposition 3.17

$(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz rings satisfy the Köthe’s conjecture.

Proof

Note that if $Nil^{*}(R)=0$, then R is semiprime and Proposition 3.16 implies that R has no nonzero nil one-sided ideal. $\square$

Proposition 3.18

If A is a skew PBW extension of a ring R which is semiprime, $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then A has no nonzero nil one-sided ideal.

Proof

Again, we proceed by contradiction. Let I be a nil right ideal of A and consider a nonzero element f of A given by $f=\sum _{i=0}^{m} a_iX_i$. Note that for every element r of R, $fr\in I$ and so $(fr)^{l}=0$, for some $l\in {\mathbb {N}}$. From Lemma 3.4, $(a_mr)^{l} = 0$, which shows that $a_mR$ is a nil right ideal of R, i.e., $a_mR=0$ (Proposition 3.16). If we repeat this argument we obtain that $f=0$, so the assertion follows. $\square$

Proposition 3.19

If A is a skew PBW extension of a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring, then A satisfies the Kothë’s conjecture.

Proof

Consider $Nil^{*}(A)=0$. Then A is semiprime, and by Remark 3.14, R is also semiprime. Since R is $\varSigma$-skew Armendariz, Proposition 3.18 asserts that the skew PBW extension A has no nonzero nil one-sided ideal, or in other words, the Köthe’s conjecture is true for A. $\square$

Example 3.20

From [16], Corollary 2.14, we know that the Hilbert basis theorem is also valid for bijective skew PBW extensions: if A is a bijective skew PBW extension of a left Noetherian ring R, then A is also a left Noetherian ring (similarly for the right version). In this way, for bijective skew PBW extensions over Noetherian rings the Köthe’s conjecture is true. With this in mind, the importance of Proposition 3.19 is precisely that we establish that the conjecture is also valid for skew PBW extensions which are not Noetherian (if R is Noetherian but some of the injective endomorphisms $\sigma _i\in \varSigma$ is not bijective then A is not necessarily Noetherian, see [16] for more details). Let us see some situations where Proposition 3.19 can be applied: (i) As we said in Example 2.13, since skew PBW extensions over domains are also domains, and every domain is a reduced ring, all these extensions are reduced rings and hence their coefficients ring is reduced, $\varSigma$-skew Armendariz, $(\varSigma ,\varDelta )$-compatible and $\varSigma$-rigid. (ii) If A is a skew PBW extension of a ring R where the coefficients commute with the variables, that is, $x_ir = rx_i$, for every $r\in R$ and each $i=1,\cdots , n$, or equivalently, $\sigma _i = {\mathrm{id}}_R$ and $\delta _i = 0$, for every i (these extensions were called constant by the authors in [37]), then it is clear that R is $(\varSigma ,\varDelta )$-compatible. So, if R is reduced, then R will be $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz whence a skew PBW extension over A will satisfy the Kothë’s conjecture. Remarkable examples of constant skew PBW extensions are the PBW extensions defined by Bell and Goodearl [5], solvable polynomial rings introduced by Kandri-Rody and Weispfenning [10], diffusion algebras [8, 9], some examples of G-algebras (see [31] for a detailed reference), some operator algebras, some quantizations of the Weyl algebra, and others (in [37] we find a list of constant skew PBW extensions).

The next theorem establishes a property for right and left annihilators on $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz rings.

Theorem 3.21

If A is a skew PBW extension of a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then

• (1) $\varphi :\{r_R(U)\mid U\subseteq R\}\rightarrow \{r_A(U)\mid U\subseteq A\}$; $C\rightarrow CA$ is bijective.

• (2) $\psi :\{l_R(U)\mid U\subseteq R\}\rightarrow \{l_A(U)\mid U\subseteq A\}$; $D\mapsto AD$ is bijective.

Proof

(1) Let $\varphi :\{r_R(U)\mid U\subseteq R\}\rightarrow \{r_A(U)\mid U\subseteq A\}$ defined by $C\rightarrow CA$, for every $C\in \{r_R(U)\mid U\subseteq R\}$ and $\varphi ':\{r_A(U)\mid U\subseteq A\}\rightarrow \{r_R(U)\mid U\subseteq R\}$ defined by $B\mapsto B\cap R$, for all $B\in \{r_A(U)\mid U\subseteq A\}$. Since $r_R(U)A=r_A(U)$, for every $U\subseteq R$, $\varphi$ is well defined. By the assumption of $(\varSigma ,\varDelta )$-compatibility of R, it follows that $r_A(V)\cap R = r_R(V_0)$, for every $V\subseteq A$, considering $V_0$ as the set of coefficients of all elements of V. This fact guarantees that $\varphi '$ is well defined. In this way, $\varphi '\varphi = {\mathrm{id}}_R$, and so $\varphi$ is injective. Now, if $B\in \{r_A(U)\mid U\subseteq A\}$, then $B=r_A(J)$, for some $J\subseteq A$. If $B_1$ and $J_1$ are the sets of coefficients of elements of B and J, respectively, let us see that $r_R(J_1)=B_1R$. Let $f=\sum _{i=0}^{m} a_iX_i\in J$ and $g=\sum _{j=0}^{t} b_jY_j\in B$. Then $fg=0$, and using that R is $\varSigma$-skew Armendariz and $(\varSigma ,\varDelta )$-compatible, $a_ib_j=0$, for every $a_i$ and $b_j$ (Proposition 3.2). Hence $J_1B_1=0$, and so $B_1\subseteq r_R(J_1)$. The $(\varSigma ,\varDelta )$-compatibility of R implies that $r_R(J_1)\subseteq B_1R$ whence $r_R(J_1)=B_1R$, i.e., $r_A(J)=B_1A$ which shows that $\varphi$ is surjective. Therefore $\varphi$ is bijective.

Similarly, we can prove (2). $\square$

From Theorem 3.21, we have immediately the following result.

Corollary 3.22

If A is a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then R satisfies the ascending chain condition on right (left) annihilators if and only if so does A.

Lemma 3.23

If R is a semiprime $(\varSigma ,\varDelta )$-compatible ring, then every annihilator $(\varSigma ,\varDelta )$-prime ideal P of R is a minimal $(\varSigma ,\varDelta )$-prime ideal of R.

Proof

Consider $P'$ a $(\varSigma ,\varDelta )$-prime ideal of R with $P'\subseteq P$. If $l_R(P)\subseteq P'$ holds, then $l_R(P)\subseteq P$, and so $l_R(P)l_R(P)=0$. Using that R is semiprime, $l_R(P)=0$, a contradiction, so $l_R(P)\nsubseteq P'$. Nevertheless, $l_R(P)P=0$ and P is a $(\varSigma ,\varDelta )$-ideal of R. By the $(\varSigma ,\varDelta )$-compatibility of R, $l_R(P)$ is a $(\varSigma ,\varDelta )$-ideal of R, and hence $P\subseteq P'$, that is, $P=P'$. $\square$

Lemma 3.24

Let A be a skew PBW extension over a ring R which is semiprime, $(\varSigma ,\varDelta )$-compatible, $\varSigma$-skew Armendariz and satisfies the ascending chain condition on right annihilators. If P is a minimal prime ideal of A, then $P\cap R$ is a minimal $(\varSigma ,\varDelta )$-prime ideal of R.

Proof

From Theorem 3.21 and Remark 3.14, we know that A is semiprime and has the ascending chain condition on right annihilators. By [18], Section 2.2.14, $P=r_A(U)$ for some subset U of A. Consider the set V consisting of all coefficients of elements of U. Using the $(\varSigma ,\varDelta )$-compatibility of R, we obtain $P\cap R=r_R(V)$. Now, for every $a\in P\cap R$ and $v\in V$, $av=0$ and hence $\sigma ^{\theta }(a)v = \delta ^{\theta }(a)v=0$, for every $\theta \in {\mathbb {N}}$ (Proposition 2.14), which shows that $P\cap R$ is a $(\varSigma ,\varDelta )$-ideal of R. If $IJ\subseteq P\cap R$, for a pair of $(\varSigma ,\varDelta )$-ideals I, J of R, then $IAJA\subseteq P$ and hence $I\subseteq P\cap R$ whence $I\subseteq P\cap R$ or $J\subseteq P\cap R$. This fact means that $P\cap R$ is an annihilator $(\varSigma ,\varDelta )$-prime ideal of R, and Lemma 3.23 guarantees the result. $\square$

Proposition 3.25

If R is a semiprime $(\varSigma ,\varDelta )$-compatible ring with the ascending chain condition on right annihilators, then every minimal prime ideal of R is a minimal $(\varSigma ,\varDelta )$-prime ideal of R and vice versa.

Proof

From [18], Section 2.2.14, we know that a prime ideal of R is minimal if and only if it is annihilator ideal. Let P be a minimal prime ideal of R and $P=r_R(U)$, for some subset U of R. Then $UP=0$, which means that for every element $r\in P$ and each $u\in U$, $ur=0$. By assumption R is $(\varSigma ,\varDelta )$-compatible, so $u\sigma ^{\theta }(r)=u\delta ^{\theta }(r)=0$, for any $\theta \in {\mathbb {N}}^{n}$ (Proposition 2.14), so $U\delta ^{\theta }(r) = U\sigma ^{\theta }(r)=0$, that is, P is $(\varSigma ,\varDelta )$-prime and so P is a minimal $(\varSigma ,\varDelta )$-prime ideal of R (Lemma 3.23).

Conversely, let P be a minimal $(\varSigma ,\varDelta )$-prime ideal of R. We know that $rad(P)=\bigcap Q_i$, where $Q_i$ is a minimal prime ideal of R with $P\subseteq Q_i$, for every i. It is clear that $P\subseteq rad(P)\subseteq Q_i$, for each i, and since $Q_i$ is a minimal prime ideal of R, it is also a minimal $(\varSigma ,\varDelta )$-prime, as we saw above. Therefore $P=Q_i$ and so $rad(P)=P$, which proves that P is a minimal prime ideal of R. $\square$

Proposition 3.26

Let A be a skew PBW extension over a ring R which is semiprime, $(\varSigma ,\varDelta )$-compatible, $\varSigma$-skew Armendariz and satisfies the ascending chain condition on right annihilators. If P is a minimal $(\varSigma ,\varDelta )$-prime ideal of R, then PA is a minimal prime ideal of A.

Proof

Let P be a minimal $(\varSigma ,\varDelta )$-prime ideal of R. Using Proposition 3.25, we have that P is a minimal prime ideal of R. One can prove that PA is a prime ideal of A. If Q is a minimal prime ideal of A with $Q\subseteq PA$, by Proposition 3.24, $Q\cap R$ is a minimal $(\varSigma ,\varDelta )$-prime ideal of R and $Q\cap R\subseteq P$, whence $Q\cap R=P$. Therefore, $Q=(Q\cap R)A = PA$ which concludes the proof. $\square$

The last result of this section is the following theorem.

Theorem 3.27

If A is a skew PBW extension over a ring R which is semiprime, $(\varSigma ,\varDelta )$-compatible, $\varSigma$-skew Armendariz and satisfies the ascending chain condition on right annihilators, then A has finitely many minimal prime ideals $Q_1,\cdots , Q_m$ with $Q_1\cdots Q_m=0$, and $Q_i=p_iA = q_iA$, for each i, where $\{p_1,\cdots , p_m\}$ is the set of all minimal prime ideals of R and $\{q_1,\cdots , q_m\}$ is the set of all minimal $(\varSigma ,\varDelta )$-prime ideals of R.

Proof

Let R be a semiprime ring which satisfies the ascending chain condition on right annihilators. Then R has finitely many minimal prime ideals $p_1,\cdots , p_m$ ([18], Section 2.2.15). From Remark 3.14 and Theorem 3.21, we know that A is semiprime and has the ascending chain condition on right annihilators. Again, by [18], Section 2.2.15, A has finitely many minimal prime ideals $Q_1,\cdots , Q_t$, say, with $Q_1Q_2\cdots Q_t=0$. Note that if P is a minimal prime ideal of R, then P is a minimal $(\varSigma ,\varDelta )$-prime ideal of R (Proposition 3.25), and PA is a minimal prime ideal of A (Proposition 3.26). In this way, if Q is a minimal prime ideal of A, then $Q\cap R$ is a minimal prime ideal of R (Lemma 3.24). Therefore $m=t$ and Proposition 3.25 implies the theorem. $\square$

4 2-Primal, Symmetric, Reversible, $ZC_n$ and IFP Properties

Recall the following definitions for a ring B: B is called (i) symmetric, if for all $a, b, c\in B$, $abc=0 \Rightarrow acb=0$; (ii) reversible, if for all $a, b\in B$, $ab = 0\Rightarrow ba=0$; (iii) B has the insertion of factors property IFP, if $ab=0$ implies that $aBb=0$, for all $a, b\in B$; (iv) reduced, if B has no nonzero nilpotent elements; (v) 2-primal, if $Nil_{*}(B) = Nil(B)$. The following relations are well known: reduced $\Rightarrow$ symmetric $\Rightarrow$ reversible $\Rightarrow$ 2-primal, but the converses are not true in general (see [17] for more details). Assuming that R is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, we show the transfer of these ring-theoretical properties between the ring of coefficients R and a skew PBW extension A over R.

Theorem 4.1

If A is a skew PBW extension over a $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz ring R, then R is 2-primal if and only if A is 2-primal.

Proof

Consider an element $f=\sum _{i=0}^{m} a_iX_i$ of Nil(A). Then $f^{l}=0$, for some $l\in {\mathbb {N}}$. From Lemma 3.4, we know that $a_i^{l}=0$, for every $0\le i\le m$. This fact means that $a_i\in Nil(R)$, for every i. By assumption R is 2-primal, so $a_i\in Nil_{*}(R)$, for each value of i, and hence $f\in Nil_{*}(R)A = Nil_{*}(A)$ (Theorem 3.12). Therefore A is 2-primal. $\square$

A ring B is called Dedekind finite, if $ab=1$ implies $ba=1$, for every $a, b \in R$. The next result says us how to obtain Dedekind finite rings.

Lemma 4.2

If R is a weak $\varSigma$-skew Armendariz ring, then R is Dedekind finite.

Proof

Consider two elements a, b of R with $ab=1$. It is clear that $(ba)^{2}=baba=ba$ which means that ba is an idempotent element of R, so it is a central element of R, which allows us to consider the equalities $1=a(ba)b = (ab) (ba) = ba$. $\square$

A ring B is called locally finite, if every finite subset in B generates a finite semigroup multiplicatively. One of the reviewers note us that the common definition of locally finite means that any finite subset generates a finite subring. If the additive group of R is periodic, then these two notions coincide. We thank to the reviewer for this remark. Of course, finite rings are locally finite.

Theorem 4.3

If R is a locally finite $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz, then $N_0(R)=Nil_{*}(R) = L-rad(R) = Nil^{*}(R) = Nil(R)$.

Proof

We follow the ideas presented in [19], Theorem 3.4. Consider R a locally finite, $(\varSigma ,\varDelta )$-compatible and weak $\varSigma$-skew Armendariz ring. We know that $N_0(R) = Nil_{*}(R) = L-rad(R) = Nil^{*}(R)$ (Theorem 3.8) which guarantees that it is enough to prove that $Nil^{*}(R)=Nil(R)$. First of all, note that for every element r of R, there exists $m_r\in {\mathbb {N}}$ with $r^{m_r}\in Z(R)$. Using that R is locally finite, there exist elements $m, k \in {\mathbb {N}}$ satisfying $r^{m} = r^{m+k}$, or equivalently, $r^{m} = r^{m}r^{k} = r^{m}r^{2k} = \cdots = r^{m}r^{mk} = r^{m(k+1)}$, i.e., $r^{km} = r^{(k-1)m}r^{m}=r^{(k-1)m}r^{m(k+1)} = r^{2mk} = (r^{km})^{2}$, which means that $r^{km}$ is an idempotent element of R. From [28], Proposition 3.9 we know that R is Abelian, so $r^{km}\in Z(R)$.

Now, take an element a of Nil(R), so $a^{l}=0$, for some $l\in {\mathbb {N}}$. With the aim of showing that $a\in Nil^{*}(R)$, we will prove that $\langle a\rangle$ is a nil ideal of R. For elements $r, s \in R$, let us see that sar is nilpotent. Note that there exists $m\in {\mathbb {Z}}$ with $(rs)^{m}\in Z(R)$. Using that $a^{l}=0$, we have $a(rs)^{m}a^{l-1}=0$ and so $a(rs)(rs)^{m-1}a^{l-1}=0$. The fact $a^{l}=0$ together with the assumptions on R and Proposition 3.6 imply $a(rs)a(rs)^{m-1}a^{l-1}=0$. Hence, $a(rs)a(rs)(rs)^{m-2}a^{l-1}=0$. Proposition 3.6 allows us to conclude the equality $a(rs)a(rs)a(rs)^{m-2}a^{l-1}=0$. If we repeat this process, we obtain $a(rs)a(rs)a\cdots a(rs)a^{l-1}=0$ whence $(ars)^{m}a^{l-1}=0$. Note that $(ars)^{m}a(rs)^{m}a^{l-2}=0$, since $(rs)^{m}\in Z(R)$. Repeating this argument l times we obtain $(ars)^{m}(ars)^{m}\cdots (ars)^{m}=0$ which imply $(ars)^{m(l-1)+1} = 0$, that is, $s(ars)^{m(l-1) + 1}ar=0$. This fact shows that $(sar)^{m(l-1)+2} = 0$ which concludes the proof. $\square$

From Theorem 4.3, we observe that if R is locally finite, $(\varSigma ,\varDelta )$-compatible, and weak $\varSigma$-Armendariz, then R is 2-primal.

Proposition 4.4

If R is locally finite, $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then $N_0(A)=Nil_{*}(A) = L-rad(A) = Nil^{*}(A) = Nil(A)$.

Proof

Theorem 3.8 says us it is enough to prove that $Nil(A)\subseteq Nil^{*}(A)$. With this in mind, consider an element $f\in Nil(A)$ given by $f=\sum _{i=0}^{m} a_iX_i$. Then $f^{l}=0$, for some $l\in {\mathbb {N}}$. From Proposition 3.4, we know that $a_i^{l}=0$, for every i, which means that $a_i\in Nil(R)$ and from Theorem 4.3, $a_i\in Nil^{*}(R)$, so Theorem 3.12 establishes that $f\in Nil^{*}(R)A=Nil^{*}(A)$, which concludes the proof. $\square$

Huh et al. showed that there is a ring B with IFP such that the polynomial ring B[x] does not satisfy this property. Nevertheless, under adequate assumptions on the ring of coefficients R, we can prove that this property is extended from R to a skew PBW extension over R.

Proposition 4.5

If A is a skew PBW extension of a ring R which is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then R has IFP if and only if A has IFP.

Proof

Let R be a IFP ring and consider two elements $f, g\in A$ given by $f=\sum _{i=0}^{m} a_iX_i$ and $g=\sum _{j=0}^{t} b_jY_j$ with $fg=0$. Proposition 3.2 establishes that $a_ib_j=0$, for every i, j. Since R has IFP, for each element r of R, $a_irb_j=0$, and so using Proposition 2.14 we obtain $fhg=0$, for all $h\in A$, which means that A has IFP. $\square$

Proposition 4.6

If A is a skew PBW extension of a ring R which is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then R is reversible if and only if A so is.

Proof

If R is a reversible ring with $fg=0$, for elements $f=\sum _{i=0}a_iX_i,\ g=\sum _{j=0}b_jY_j$, Proposition 3.2 implies that $a_ib_j=0$, for every i, j, and hence $b_ja_i=0$ whence $gf=0$, again by Proposition 3.2. $\square$

Proposition 4.7

If A is a skew PBW extension of a ring R which is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then R is symmetric if and only if A so is.

Proof

Consider R a symmetric ring and f, g, h elements of A with $fgh=0$, given by $f=\sum _{i=0}^{m} a_iX_i, g=\sum _{j=0}^{t} b_jY_j$ and $h=\sum _{k=0}^{s}c_kZ_k$. From Proposition 3.4, we have that $a_ib_jc_k=0$, for every i, j, k, and using that R is symmetric, $a_ic_kb_j=0$. Proposition 2.14 guarantees that $fhg = 0$. $\square$

Following Anderson and Camillo [2], we say that a ring B satisfies the property $ZC_l$, if for elements $a_1,a_2\cdots ,a_l\in B$ with $a_1a_2\cdots a_l=0$, then we have $a_{\gamma (1)}a_{\gamma (2)}\cdots a_{\gamma (l)} = 0$, for every $\gamma \in {\mathrm{Sym}}(l),\ l\ge 2$, where ${\mathrm{Sym}}(l)$ denote the permutation group on l letters.

Proposition 4.8

If A is a skew PBW extension of a ring R which is $(\varSigma ,\varDelta )$-compatible and $\varSigma$-skew Armendariz, then R satisfies $ZC_l$ if and only if A satisfies $ZC_l$.

Proof

Suppose that R satisfies $ZC_l$ and $f_1f_2\cdots f_l=0$, for elements $f_1,\cdots , f_l\in A$. From the assumptions on R and Lemma  3.4 we know that $a_1a_2\cdots a_l = 0$, for every element $a_i\in C_{f_i}$, $1\le i\le l$. Using that R satisfies $ZC_l$, for every $\gamma \in {\mathrm{Sym}}(l)$, we have that $a_{\gamma (1)}a_{\gamma (2)}\cdots a_{\gamma (l)} = 0$. The facts established in Proposition 2.14 allows us to conclude that $f_{\gamma (1)}f_{\gamma (2)}\cdots f_{\gamma (l)} = 0$. $\square$