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

Armando Reyes,Héctor Suárez

doi:10.33044/revuma.v59n1a08

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).

Highlights

In [22], Kaplansky defined a ring B as a Baer ring, if the right annihilator of every nonempty subset of B is generated by an idempotent
A ring B is called right p.p., if the right annihilator of each element of B is generated by an idempotent (or equivalently, rings in which each principal right ideal is projective)
Birkenmeier et al [8] defined a ring to be called a right principally quasi-Baer (or right p.q.-Baer) ring, if the right annihilator of each principal right ideal of B is generated by an idempotent

Summary

Introduction

In [22], Kaplansky defined a ring B as a Baer (resp. quasi-Baer, which was defined by Clark [10]) ring, if the right annihilator of every nonempty subset (resp. ideal) of B is generated by an idempotent (the objective of these rings is to abstract various properties of von Neumann algebras and complete ∗-regular rings; Clark used the quasi-Baer concept to characterize when a finite-dimensional algebra with unity over an algebraically closed field is isomorphic to a twisted matrix units semigroup algebra). With all these works in mind, our second approach to a notion of Armendariz ring of skew PBW extensions consists in establishing the conditions (SA1) and (SQA1) for the case of skew PBW extensions with the aim of generalizing the results presented in [35] about Σ-rigid rings, and the results presented in [15] and [16] for Ore extensions of injective type. In. Section 4, we introduce the notions (SA1) and (SQA1) for skew PBW extensions (Definitions 4.1 and 4.11, respectively) with the aim of generalizing the results presented in [35] about Baer, quasi-Baer, p.p. and p.q.-Baer rings for Σ-rigid rings, to the more general setting of (Σ, ∆)-compatible rings, see Theorems 4.2, 4.4, 4.7, and 4.15. [12, 32, 27, 33, 34, 26, 35, 36, 38])

Skew PBW extensions

Examples