Generalized π-Baer rings

Abstract

We call a ring R generalized right π-Baer, if for any projection invariant left ideal Y of R , the right annihilator of Y n is generated, as a right ideal, by an idempotent, for some positive integer n , depending on Y. In this paper, we investigate connections between the generalized π-Baer rings and related classes of rings (e.g., π-Baer, generalized Baer, generalized quasi-Baer, etc.) In fact, generalized right π-Baer rings are special cases of generalized right quasi-Baer rings and also are a generalization of π-Baer and generalized right Baer rings. The behavior of the generalized right π-Baer condition is investigated with respect to various constructions and extensions. For example, the trivial extension of a generalized right π-Baer ring and the full matrix ring over a generalized right π-Baer ring are characterized. Also, we show that this notion is well-behaved with respect to certain triangular matrix extensions. In contrast to generalized right Baer rings, it is shown that the generalized right π-Baer condition is preserved by various polynomial extensions without any additional requirements. Examples are provided to illustrate and delimit our results.