Principally quasi-Baer modules and their generalizations

The purpose of this article is to further the study of principally quasi-Baer modules and -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 -principally quasi-Baer (simply, -p.q.-Baer) modules. Examples which show that the notion of an -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 -p.q.-Baer module inherits the property. Furthermore, we obtain that every direct sum of copies of an -p.q.-Baer module is an -p.q.-Baer module. We provide conditions when (-)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.