π-endo Baer modules

Abstract

Let N be a submodule of a right R-module M_R, and $\mathbf{H}=\mathit{End}(M_R).$ Then N is said to be projection invariant in M, denoted by $N{⊴}_{p}M,$ if $eN\subseteq N$ for all $e=e^2\in \mathbf{H}.$ We call M_R π-endo Baer, denoted π-e.Baer, if for each $N{⊴}_{p}M$ there exists $e=e^2\in \mathbf{H}$ such that $l_{\mathbf{H}}(N)=\mathbf{H}e$ where $l_{\mathbf{H}}(N)$ denotes the left annihilator of N in H. We show that this class of modules lies strictly between the classes of Baer and quasi-Baer modules introduced in 2004 by Rizvi and Roman. Several structural properties are developed. In contrast to the Baer modules of Rizvi and Roman, the free modules of a Baer ring are π-e.Baer. Moreover, (co-) nonsingularity conditions are introduced which enable us to extend the Chatters-Khuri result (connecting the extending and Baer conditions in a ring) to modules. We provide examples to illustrate and delimit our results.

Keywords:

• Baer module
• endomorphism rings
• projection invariant submodule
• quasi-Baer module
• π-extending module
• π-e.Baer module

2010 AMS Subject Classification:

• Primary: 16D10
• 16S50
• Secondary: 16D80

Acknowledgments

The second author was supported by The Scientific and Technological Research Council of Turkey, TUBITAK (BIDEB-2219) and this work was carried out during her visit to the University of Louisiana at Lafayette.