We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew P BW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew P BW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as P BW extensions, well known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.

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....

In this paper we present the notion of skew Π-Armendariz for the non-commutative rings known as σ-PBW extensions. This concept generalizes several definitions of Armendariz rings presented in the literature for these extensions, and in particular, for Ore extensions of injective type. We investigate the relations between skew Π-Armendariz, Σ-rigid, (Σ, ∆)-skew Armendariz and Σ-skew Armendariz rings rings.

Abstract. The aim of this paper is to study skew Poincaré-Birkhoff-Witt extensions of Baer, quasi-Baer, p.p. and p.q.-Baer rings. Using a notion of rigidness, we prove that these properties are stable over this kind of extensions.

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