A CLASSIFICATION OF RING ELEMENTS IN SKEW PBW EXTENSIONS OVER COMPATIBLE RINGS

For a skew PBW extension over a right duo compatible ring, we characterize several kinds of their elements such as units, idempotent, von Neumann regular, π-regular and the clean elements. As a consequence of our treatment, we extend several results in the literature for Ore extensions and


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With all above results in mind, our purpose in this paper is to establish analogue characterizations to the established in [21] for Ore extensions but now for a more general kind of noncommutative rings. We are taking about the skew PBW extensions, which are noncommutative rings of polynomial type more general than Ore extensions of injective type (i.e., when σ is injective). These extensions were introduced by Gallego and Lezama [18] with the aim of extending the PBW extensions introduced by Bell and Goodearl [11]. During the last years, several authors have been studying ring and module theoretical properties of these objects (e.g., [5], [6], [22], [23], [24], [38], [41], [47], [51], [56], [60] and [64]). In Section 2 we will say some words about the relations between skew PBW extensions and other families of noncommutative rings of polynomial type considered in the literature. are well-known: reduced ⇒ reversible ⇒ semicommutative ⇒ 2-primal ⇒ weakly 2-primal ⇒ NI, but the converses do not hold (see [16] and [34]). A ring B is said to be right (respectively, left) duo, if every right (respectively, left) ideal is an ideal.
The importance of the study of all these classes of rings is due to their importance in the Köthe's conjecture (see [10] and [42]).
To finish this introduction, we describe the structure of the article. In Section 2 we recall some useful results about skew PBW extensions for the rest of the paper. In Section 3 we establish key facts about (Σ, ∆)-compatible rings which are important in the proofs of the results obtained in the following sections. Precisely, in Section 4 we characterize the units of a skew PBW extension over a right duo (Σ, ∆)compatible ring, while in Section 5 we establish relations between the idempotent, von Neumann regular and local, and clean elements of a right duo (Σ, ∆)-compatible ring R and those elements corresponding of a skew PBW extension A over R. The results obtained in Sections 4 and 5 generalize corresponding results presented by 78 MARYAM HAMIDIZADEH, EBRAHIM HASHEMI AND ARMANDO REYES the first two authors in [21] for Ore extensions of injective type. We have to say that the techniques used here are fairly standard and follow the same path as other text on the subject, and hence the results presented here are new for skew PBW extensions and all they are similar to others existing in the literature. Our paper can be considered as a modest contribution to the study of ring elements of noncommutative rings of polynomial type which can not be expressed as Ore extensions but as skew PBW extensions. Finally, in the future work, we consider a possible topic of research concerning modules over skew PBW extensions.

Skew PBW extensions
Skew PBW extensions are a direct generalization of PBW extensions introduced by Bell and Goodearl [11]. They also are strictly more general than Ore extensions of injective type (see [58], Example 1, for a list of noncommutative rings which are skew PBW extensions but not Ore extensions). Nevertheless, as time went by, we and others realized that these extensions also generalize several families of noncommutative rings appearing in representation theory, Hopf algebras, quantum groups, noncommutative algebraic geometry and other algebras of interest in the context of theoretical physics (e.g., [39] and [54] for more details). Next, we mention briefly some of them: (1) Universal enveloping algebras of finite dimensional Lie algebras. (2) Almost normalizing extensions defined by McConnell and Robson [43]. (3) Solvable polynomial rings introduced by Kandri-Rody and Weispfenning [31]. (4) Diffusion algebras studied by Isaev, Pyatov, and Rittenberg [30]. (5) 3-dimensional skew polynomial algebras studied by Rosenberg [61] (see also [55]).
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 (see Definition 2.1). In fact, the skew PBW extensions share examples of algebras with generalized Weyl algebras defined by Bavula [8] (also known as hyperbolic algebras defined by Rosenberg [61]), with G-algebras introduced by Apel [4] and some PBW algebras defined by Bueso et al., [15], (both G-algebras and PBW algebras take coefficients in fields and assume that coefficients commute with variables), Auslander-Gorenstein rings, some Calabi-Yau and skew Calabi-Yau algebras, some Artin-Schelter regular algebras, some Koszul and augmented Koszul algebras, quantum polynomials, some quantum universal enveloping algebras, some graded skew Clifford algebras and others (e.g., [13], [39], [63] and [64]). As we can see, skew PBW extensions include a considerable number of noncommutative rings 79 of polynomial type, so a classification of ring elements of these extensions will establish results for algebras not considered before and, of course, it will cover also several treatments in the literature. Definition 1). Let R and A be rings. We say that A is a skew PBW extension of R (also called a σ-PBW extension of R), which is denoted by A := σ(R) x 1 , . . . , x n , if the following conditions hold: (i) R is a subring of A sharing the same multiplicative identity element.
(ii) Given the importance of monomial orders in the proofs of the results presented in Section 4, next we recall some key facts about these for skew PBW extensions.
Let be a total order defined on Mon(A). If x α x β but x α = x β , we will write x α x β . If f is a nonzero element of A, then f can be expressed uniquely as f = a 0 + a 1 X 1 + · · · + a m X m , where every X i is a monomial with a i ∈ R, and X m · · · X 1 (eventually, we will use expressions as f = a 0 + a 1 Y 1 + · · · + a m Y m , with a i ∈ R, and Y m · · · Y 1 ). With this notation, we define lm ( Following [18], Definition 11, if is a total order on Mon(A), we say that is a monomial order on Mon(A), if the following conditions hold: lm(x γ x α x λ ) (the total order is compatible with multiplication).
In [18], monomial orders are also called admissible orders. The condition The importance of considering monomial orders on Mon(A) can be appreciated in [18] where the Gröbner theory for left ideals of skew PBW extensions was studied.
The result established in [15], Chapter 2 or [14], Theorem 1.2, for PBW rings in the sense of [15] motivated the following result for skew PBW extensions. Connections with filtered rings and their corresponding graded rings can be found in [39]. (i) For each x α ∈ Mon(A) and every 0 = r ∈ R, there exist unique elements Remark 2.6. With respect to the Proposition 2.5, we have two observations: (i) ( [49], Proposition 2.9) If α := (α 1 , . . . , α n ) ∈ N n and r ∈ R, then (ii) ( [49], Remark 2.10) Using (i), it follows that for the product In this way, when we compute every summand of a i X i b j Y j we obtain products of the coefficient a i with several evaluations of b j in σ's and δ's de- Several examples of skew PBW extensions can be found in [39] and [59].
To finish this section, we include one more definition and a result about quotient rings of skew PBW extensions. (2) If I is proper and σ i (I) = I, for every 1 ≤ i ≤ n, then A/IA is a skew PBW extension of R/I. Moreover, if A is bijective, then A/IA is bijective.
(3) Let R be left (right) Noetherian and σ i bijective, for every 1 ≤ i ≤ n. Then σ i (I) = I, for every i and IA = AI. If I is proper and A is bijective, then A/IA is a bijective skew PBW extension of R/I.

(Σ, ∆)-compatible rings
Following Krempa [35], an endomorphism σ of a ring B is said to be rigid, if It is clear that any rigid endomorphism of a ring is a monomorphism, and σ-rigid rings are reduced ( [28], p. 218). Properties of σ-rigid rings have been studied by several authors (c.f. [35] and [28]). With this In this case, the endomorphism σ is injective. Since one can appreciate the relation between these notions and σ-rigid rings, in [25], Lemma 2.2, it was shown that a ring B is (σ, δ)-compatible and reduced if and only if B is σ-rigid.
Hence σ-compatible rings generalize σ-rigid rings for the case B is not assumed to be reduced. The natural task for us is to extend this notion of compatibility to a more general context of Ore extensions of injective type, that is, the family of skew PBW extensions; this is precisely the content of Definition 3.2. Before, we recall the notion of Σ-rigid ring introduced by the third author.  Note that if Σ is a rigid endomorphisms family, then every element σ i ∈ Σ is a monomorphism. In fact, Σ-rigid rings are reduced rings: if B is a Σ-rigid ring and r 2 = 0 for r ∈ B, then we obtain the equalities 0 = rσ α (r 2 )σ α (σ α (r)) = rσ α (r)σ α (r)σ α (σ α (r)) = rσ α (r)σ α (rσ α (r)), i.e., rσ α (r) = 0 and so r = 0, that is, B is reduced (note that there exists an endomorphism of a reduced ring which is not a rigid endomorphism, see [28], Example 9). Σ-rigid rings have been investigated in several papers (e.g., [47], [50], [58] and [59]).   enveloping algebra of a Lie algebra, and others); 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 class of diffusion algebras; Weyl algebras; additive analogue of the Weyl algebra; multiplicative analogue of the Weyl algebra; some quantum Weyl algebras as A 2 (J a,b ); the quantum algebra U (so(3, k)); the family of 3-dimensional skew polynomial algebras (there are exactly fifteen of these algebras, see [55]); Dispin algebra U (osp(1, 2)); Woronowicz algebra W v (sl(2, k)); the complex algebra V q (sl 3 (C)); q-Heisenberg algebra H n (q); the Hayashi algebra W q (J), and more. 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 q (S q ) on a quantum space S q , and more.
(c) Several algebras of quantum 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. See [54] or [59] for a detailed list of examples.
The following example illustrates that the converse of Proposition 3.4 is false.
Then δ is a σ-derivation of B 3 , and we have the following facts: B 3 is a (σ, δ)compatible ring, (ii) B 3 is not σ-rigid.
Nevertheless, Proposition 3.7 shows the importance of reduced rings in the equivalence of both families of rings.  If f = a 0 + a 1 X 1 + · · · + a m X m ∈ A, r ∈ R, and f r = 0, then a i r = 0, for every i.
For the next proposition, we assume that the elements d i,j ∈ R in Definition 2.1 (iv) are central in R. Remark 3.11. The notion of compatibility has been very useful in the study of different ring theoretical properties of skew PBW extensions, for example see [51], [52], [59] and [60].

Units
In this section we characterize the units of a skew PBW extension over a right duo (Σ, ∆)-compatible ring (see [19], [25], [52] and [56] for some classes of rings which satisfy these conditions in the context of Ore extensions and skew PBW extensions, respectively). With this in mind, we establish analogue results to the obtained by the first two authors in [21] for the case of Ore extensions.
Again, the induction hypothesis guarantees that a 0 b 0 = c and a i b j = 0, for 1 ≤ i + j ≤ m + t, which concludes the proof.  cb t = f gb t = (a 0 + a 1 X 1 + · · · + a m−1 X m−1 )gb t , by the semicommutativity of R and Remark 2.6 (ii). Consider two cases: If b t σ αt (b t ) = 0, then gb t = 0, and using induction hypothesis there If m ≥ 1, then there exists s ≥ 1 such that a s m g = 0.
Proof. The proof uses a similar argument to the established in the proof of Proposition 4.3.

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For the next theorem, the first important result of the paper, let I g be the right ideal of R generated by the coefficients of nonzero elements of a skew PBW extension A. b j Y j are nonzero elements of A such that f g = c ∈ R, then there exist a nonzero element r ∈ I g and an element a ∈ R with f r = ca. In the case that b 0 is a unit in R, then a 1 , a 2 , . . . , a m are nilpotent.
Proof. Note that if f = a 0 then a 0 g = c, which implies that a 0 b 0 = c and a 0 b j = 0, for every j ≥ 1. Hence, r = b 0 and a = 1. Consider f = m i=0 a i X i with i ≥ 1. Let us prove the assertion by using induction on i. If t = 0, then g = b 0 = 0. Since f g = (a 0 + a 1 X 1 + · · · + a m X m )b 0 = c, we obtain that a 0 b 0 = c and a i b 0 = 0, for every i ≥ 1 (Proposition 4.1). If this is the case, r = b 0 and a = 1 as we wish.
Suppose that t ≥ 1 and that the assertion is true for all polynomials of degree less than t. We consider two cases: Case 1: If a m X m g = 0, then a m b j = 0, for each 0 ≤ j ≤ m by Proposition 4.1, and so a m I g = 0. This means that c = f g = (a 0 + a 1 X 1 + · · · + a m−1 X m−1 )g, and so there is 0 = r ∈ I g and a 1 ∈ R such that (a 0 + a 1 X 1 + · · · + a m−1 X m−1 )r 1 = ca 1 .
Since a m X m r 1 = 0, it follows that f r 1 = ca 1 . Hence r = r 1 and a 1 = a, as we wanted.
Case 2: Let a m X m g = 0. Then there is 0 ≤ j < m such that a m b j = 0 (j is the greatest). By Proposition 4.4, there exist t ≥ 2 such that a s m b j = 0, but a s−1 m b j = 0. As R is right duo, there is b ∈ R such that a s−1 m b j = b j b. Consider g 1 = gb. Then f g 1 = f gb = cb and 0 = I g1 ⊆ I g . Now a m X m annihilates coefficients of g 1 from j-th to t-th. So, after repeating this process a finite number of times, it is concluded that there are 0 = k ∈ A and r 1 ∈ R such that deg(k) < t, f h = cr 1 and a m X m k = 0. Now, the result follows from the Case 1.
Let b 0 be a unit in R. We will prove that a 1 , a 2 , . . . , a m are all nilpotent. Since R is right duo, R is semicommutative, and so Nil(R) is an ideal of R, by [40], Lemma 3.1. Thus R = R/Nil(R) is a reduced ring. Since R is (Σ, ∆)-compatible, Nil(R) is a (Σ, ∆)-compatible ideal of R, by Proposition 3.6 (see [23], Definition 3.1 for the notion of (Σ, ∆)-compatible ideal). Hence R is Σ-rigid, as one can check using a similar reasoning to the used in [20], Proposition 2.1 and having in mind Proposition 3.7. In this way, by Proposition 2.8, A := A/IA is a skew PBW extension of R/Nil(R). Since f g = c ∈ R, it follows that f g = c in A. Thus 88 MARYAM HAMIDIZADEH, EBRAHIM HASHEMI AND ARMANDO REYES a 0 b 0 = c and a i b j = 0, for each i + j ≥ 1, by Proposition 4.1. Hence a i = 0, for each i ≥ 1, since b 0 is a unit. Therefore a i is nilpotent, for each i ≥ 1 as desired. Proof. It is well-known that a right duo ring is 2-primal, so we have that R is 2-primal. In this way, A is a 2-primal ring ( [41], Corollary 3.10). Now, since every 2-primal ring is a weakly 2-primal ring, then Nil(A) = L−rad(A). Again, using [41],  Conversely, let a 0 be a unit element and a 1 , . . . , a m be nilpotent elements of R. Proposition 4.6 shows that the element a 1 X 1 + · · · + a m X m belongs to Nil(R)A = L − rad(A), and from [36], Lemma 10.32, we know that L − rad(A) ⊆ J(A), and so a 1 X 1 +· · ·+a m X m ∈ J(A). This shows that the element f = a 0 +a 1 X 1 +· · ·+a m X m is a unit element of A and hence the proof ends.
Before we state Corollary 4.8, we consider the following: if A = σ(R) x 1 , . . . , x n , and S is a subset of R, then SA will denote the set of elements of A with coefficients in S, that is,

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Our purpose in this section is to determinate several relations between the idempotent, von Neumann regular and local, and clean elements of a right duo (Σ, ∆)compatible ring R and those elements corresponding of a skew PBW extension A = σ(R) x 1 , . . . , x n . As in Section 4, we follow the ideas presented by the first two authors in [21] for the context of Ore extensions (see also [7]).
Before, we recall that if I is a nil ideal in a ring B (i.e., I ⊆ J(B)), and a ∈ B is such that a ∈ B = B/I is an idempotent element, then there exists an idempotent element e ∈ aB with e = a ∈ B, by [36], Theorem 21.28 (e.g., [32]). Proof. Since R is a right duo and (Σ, ∆)-compatible ring, Nil(R) is a (Σ, ∆)compatible ideal of R. Hence, by [22], Lemma 3.5 or [57], Theorem 3.9, R/Nil(R) is a Σ-rigid ring. Now, by Proposition 2.8, Using that R is Σ-rigid, we obtain that R/I is Σ-skew Armendariz, by [53], Proposition 3.4. Now, having in mind that R/I is (Σ, ∆)-compatible, we can conclude that a i = 0, for each 1 ≤ i ≤ n, and a 2 0 = a 0 . It means that f = a 0 ∈ R/Nil(R). Therefore, [36], Theorem 21.28 implies that a 0 = e ∈ R/Nil(R), for some idempotent e ∈ B.
The next assertion is the third important result of the paper. Proof. By Proposition 4.9, a i ∈ Nil(R), for each 1 ≤ i ≤ n, and there is an idempotent element e ∈ R and a nilpotent element w ∈ R such that a 0 = e + w.
by Proposition 3.9. Since e is an idempotent element of R, and R is right duo (Σ, ∆)-compatible, it follows that σ i (e) = e and δ i (e) = 0, for every i = 1, . . . , n.
Hence, by a similar way as used in the proof of [22], Theorem 3.3, one can prove that f = 0 and so f = a 0 is an idempotent element of A. This completes the proof.
From results above we have immediately the following assertions.  For the next result, Theorem 4.14, which is the fourth important result of the paper, before we need the following facts about Abelian rings. Then the following statements are equivalent: (1) a ∈ vnr(B); (2) ava = a, for some v ∈ U (B); (3) a = ve, for some v ∈ U (B) and e ∈ Idem(B); Proof. By assumption R is a right duo ring, so Corollary 4.12 implies that A is an , for every j ≥ 0, for some u ∈ U (R) and e ∈ Idem(R)    (Σ, ∆)-compatible rings can be found in [53], [54], [56] and [64]. In this way, the results obtained in Sections 4 and 5 can be illustrated with every one of these noncommutative rings. More precisely, if A is a skew PBW extension over a reduced ring R where the coefficients commute with the variables, that is, x i r = rx i , for every r ∈ R and each i = 1, . . . , n, or equivalently, σ i = id R and δ i = 0, for every i 92 MARYAM HAMIDIZADEH, EBRAHIM HASHEMI AND ARMANDO REYES (these extensions were called constant in [56], Definition 2.6 (a)), then it is clear that R is a Σ-rigid ring. Some examples of these extensions are the following: (i) PBW extensions defined by Bell and Goodearl (which include the classical commutative polynomial rings, universal enveloping algebra of a Lie algebra, and others); 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) (ii) Solvable polynomial rings introduced by Kandri-Rody and Weispfenning (iii) 3-dimensional skew polynomial algebras (e.g., [55] and Some Koszul and quadratic algebras. A detailed reference of every one of these algebras can be found in [39], [58] and [63]. Of course, we also encounter examples of skew PBW extensions which are not constant (see [39] for the definition of each one of these algebras): the quantum plane O q (k 2 ); the Jordan plane; the algebra of q-differential operators D q,h [x, y]; the mixed algebra D h ; the operator differential rings and the algebra of differential operators D q (S q ) on a quantum space S q .
Last, but not least, our results can also be applied to the noncommutative rings considered by Artamonov et al., [6].

Future work
The notion of (σ, δ)-compatibility has been considered in the study of modules over Ore extensions (e.g., [1] and [3]). For instance, in [1] the authors introduced the notions of skew-Armendariz modules and skew quasi-Armendariz modules which are generalizations of σ-Armendariz modules and extend the classes of non-reduced skew-Armendariz modules. They obtained different properties of modules over the ring of coefficients and the corresponding Ore extension. Now, recently, the notion of (Σ, ∆)-compatibility in the context of modules over skew PBW extensions has been considered by the third author in [51] with the aim of obtaining similar results to those established in [1] for Ore extensions, and also in [46] with the purpose of characterizing the associated prime ideals over these extensions generalizing the treatment developed in [3] for Ore extensions. Having this in mind and considering the results obtained in this paper, we think as a future work to investigate a classification of several types of elements in modules over these extensions.