Richart And Principally Quasi-Baer Modules

We introduce the notions of left Richart and left principally quasi-Baer property in a general module theoretic setting. Suppose M is a left R-module. If the left annihilator of any element of End_R(M) is a direct summand of M, then M is called left Richart;if the left annihilator of any principal left ideal of End_R(M) is a direct summand of M, then M is called left principally quasi-Baer. Left Richart and left principally quasi-Baer modules are the extension of left Richart rings and left principally quasi-Baer rings, respectively. In this paper, we give the equivalent characterizations of left Richart and left principally quasi-Baer modules, respectively, and show that a direct summand of a left Richart (left principally quasi-Baer) module inherits the property. As to left Richart modules, we show that every finitely generated abelian groups is left Richart exactly if and only if it is either semisimple or torsion-free, a sufficient and necessary condition for a sum of left Richart modules to be left Richart is provided. As to left principally quasi-Baer modules, we show that left principally quasi-Baer rings have Morita invariant property, and that every finitely generated projective modules over a left principally quasi-Baer ring is left principally quasi-Baer. Among other results, we also show that the endomorphism ring of a left Richart (left principally quasi-Baer) module is a left Richart (left principally quasi-Baer) ring, while the converse is not true in general. Connections between left Richart, left principally quasi-Baer and regular modules are further studied.