A notion of compatibility for Armendariz and Baer properties over skew PBW extensions

Abstract

In this paper we are interested in studying the properties of Armendariz, Baer, quasi-Baer, p.p. and p.q.-Baer over skew PBW extensions. Using a notion of compatibility, we generalize several propositions established for Ore extensions and present new results for several noncommutative rings which can not be expressed as Ore extensions (universal enveloping algebras, diffusion algebras, and others). 2010 Mathematics Subject Classification. Primary: 16S36; 16E50. Secondary: 16W60. 157 158 ARMANDO REYES AND HÉCTOR SUÁREZ in [6], they proved that a ring B is right p.q.-Baer if and only if B[x] is right p.q.-Baer. In the context of noncommutative rings, more exactly polynomial extensions known as Ore extensions B [x; σ, δ] of injective type, i.e., when σ is injective, we found several works (cf. [10,5,6,7,13,19,8,9,17,18,15,16]). Some of these works consider the case δ = 0 and σ an automorphism, or the case where σ is the identity. It is important to say that the Baerness and quasi-Baerness of a ring B and an Ore extension B[x; σ, δ] of B do not depend on each other. More exactly, there are examples which show that there exists a Baer ring B but the Ore extension B[x; σ, δ] is not right p.q.-Baer; similarly, there exist Ore extensions B[x; σ, δ] which are quasi-Baer, but B is not quasi-Baer (see [19] for more details).One of the most important kinds of rings for which all the above properties have been studied are the σ-rigid rings (cf. [24,19,18]). Following Krempa [24], an endomorphism σ of a ring B is called rigid if aσ(a) = 0 implies a = 0, for a ∈ B, and a ring B is said to be σ-rigid, if there exists a rigid endomorphism σ of B. One can see that any rigid endomorphism of a ring is a monomorphism, and σ-rigid rings are reduced rings ([19, p. 218]). With the aim of generalizing the σ-rigid rings in the context of Ore extensions, in [2] Annin introduced the notion of compatibility: a ring B is called σ-compatible if for every a, b ∈ B, we have ab = 0 if and only if aσ(b) = 0 (necessarily, the endomorphism σ is injective); B is called δ-compatible if for each a, b ∈ B, ab = 0 ⇒ aδ(b) = 0. If B is both σ-compatible and δ-compatible, B is called (σ, δ)-compatible. Note that σ-rigid rings are (σ, δ)compatible rings ([15, Lemma 3.3]), but the converse is false ([15, Examples 2.1, 2.2 and 2.3]). Nevertheless, Hashemi et al., in [16, Lemma 2.2], showed 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 where B is not assumed to be reduced. All the above properties have been also studied for (σ, δ)-compatible rings: in [16] it was imposed the (σ, δ)-compatibility on the ring B and it was proved that (i) the ring B is quasi-Baer if and only if B[x; σ, δ] is quasi-Baer; (ii) the ring B is left p.q.-Baer if and only if B[x; σ, δ] is left p.q.-Baer. In this way, the treatment in [16] is a generalization of [6, Theorem 1.8] and [9, Theorem 3.1].With all the above facts in mind, a natural question for a given class of Baer, quasi-Baer, p....

Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions

“…As we said in the Introduction, in [36] and [56] the second author considered the question about the property of being 2-primal and the minimal prime ideals for skew PBW extensions by using a different approach to the established in this paper. More exactly, there, the notions of compatible ring, skew Armendariz ring (see [49], [52] and [53] for more details) and the ascending chain condition on right annihilators (see [58]) on the ring of coefficients were key in the characterization of minimal prime ideals of these extensions. Therefore, the results obtained in this paper are another approach to the study of these ideals of skew PBW extensions.…”

“…It is important to say that this paper continues the study of ideals of these extensions initiated in [21], [30], [37], [53], [56] and [58]. We remark that in [37] and [56] the second author considered the question about the property of being 2-primal and the minimal prime ideals for skew PBW extensions by using a different approach to the presented here.…”

Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

“…Skew PBW extensions have been investigated enough, with many homological properties of them having been studied, as well as their Gröbner theory ( [8,[11][12][13][16][17][18][19][20][21][22][23][24]). We conclude this introductory section with some known results about skew PBW extensions and semi-graded rings that we use in the present paper.…”

Armendariz modules over skew PBW extensions

“…The next definition introduce a more general class of modules than those established in Definition 3.1. As it was established in [46], Proposition 3.3, the importance of (Σ, ∆)-compatible rings is that they are more general than Σ-rigid rings defined and characterized by the author in [42] in terms of the properties of being Baer, p.p., p.q., and p.q.-Baer (Σ-rigid rings are a generalization of α-rigid rings defined by Krempa in [25] and studied by Hong et. al., [18]).…”