π-Baer rings

Abstract

We say a ring [Formula: see text] is [Formula: see text]-Baer if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. In this paper, we study connections between the [Formula: see text]-Baer condition and related conditions such as the Baer, quasi-Baer and [Formula: see text]-extending conditions. The [Formula: see text]-by-[Formula: see text] generalized triangular and the [Formula: see text]-by-[Formula: see text] triangular [Formula: see text]-Baer matrix rings are characterized. Also, we prove that a [Formula: see text]-by-[Formula: see text] full matrix ring over a [Formula: see text]-Baer ring is a [Formula: see text]-Baer ring. In contrast to the Baer condition, it is shown that the [Formula: see text]-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.