Principally quasi-Baer modules and their generalizations

Abstract

The purpose of this article is to further the study of principally quasi-Baer modules and $\mathfrak{L}$-principally quasi-Baer modules as these properties play an important role in the study of quasi-Baer module. First, we provide some basic results and a characterization of a principally quasi-Baer (simply, p.q.-Baer) module in terms of its endomorphism ring by using the pq-local-retractable property. In addition, we fully characterize when a finite direct sum of arbitrary p.q.-Baer modules is p.q.-Baer. Next, we obtain characterizations and properties of $\mathfrak{L}$-principally quasi-Baer (simply, $\mathfrak{L}$-p.q.-Baer) modules. Examples which show that the notion of an $\mathfrak{L}$-p.q.-Baer module is distinct from that of a p.q.-Baer module are provided. It is shown that every direct summand of an $\mathfrak{L}$-p.q.-Baer module inherits the property. Furthermore, we obtain that every direct sum of copies of an $\mathfrak{L}$-p.q.-Baer module is an $\mathfrak{L}$-p.q.-Baer module. We provide conditions when ($\mathfrak{L}$-)p.q.-Baer modules become quasi-Baer modules. In particular, if every direct sum of copies of a module M is p.q.-Baer then the module M is a quasi-Baer module.

Keywords:

• ($\mathfrak{L}$-)p.q-Baer module
• pq-local-retractability
• quasi-Baer module

Mathematics Subject Classification 2010:

• Primary 16D40
• 16D70
• 16D80
• 16S50

Acknowledgments

The author is very thankful to the referee for the nice comments about this work and also very much appreciate a prompt and thorough report from the referee on the paper.

Additional information

Funding

This work was supported by research fund of Chungnam National University.