Virtually (dual) Baer Modules

As a generalization of Baer modules,the concept of virtually Baer modules is introduced.A module M is called virtually Baer if the right annihilator of a left ideal of the endomorphism ring of M is isomorphic to a direct summand of M.We illustrate that virtually Baer modules are proper generalization of Baer modules.Basic properties of virtually Baer modules are investigated.Under certain conditions,virtually Baer modules are closed under direct summands,the relationships between virtually Baer modules and GSIP modules、generalized SSIP modules are discussed.It is shown that ring R is a semisimple Artinian ring if and only if every right R-module is a virtually Baer module if and only if every right R-module is a generalized SSIP module.As a generalization of dual Baer modules,the concept of virtually dual Baer modules is introduced.Basic properties of virtually dual Baer modules are studied.It is shown that M is a virtually dual Baer module if and only if M is a virtually semisimple module when it is a regular module.The relationships between virtually dual Baer modules and some related modules and rings are discussed.