Essentially Baer modules

The concepts Rickart rings and Baer rings have their roots in the theory of linear operators in Hilbert space. The concept of Baer rings was introduced by I. Kaplansky in 1955 and the concept of Rickart rings was introduced by Maeda in 1960. In recent years, many authors have been actively studied the module theoretic analogue of these rings.
In this paper, we introduce the concept of essentially Baer modules, essentially quasi-Baer modules and study their properties. We prove that, every direct summand of an essentially Baer module is also an essentially Baer module. We also prove that, every free module over essentially quasi-Baer ring is an essentially quasi-Baer module and each finitely generated free module over dual essentially quasi-Baer ring is a dually essentially quasi-Baer module; if $M$ is CS-Rickart and $M$ has the SSIP-CS then $M$ is essentially Baer. The converse is true if $\mathrm{Soc}M\unlhd M$; if $M$ is d-CS-Rickart and $M$ has the SSSP-d-CS then $M$ is dual essentially Baer. The converse is true if $\mathrm{Rad}M\ll M$; if $R$ is a right semi-artinian ring, then $M$ is an essentially Baer module if and only if $M$ is CS-Rickart and $M$ has the SSIP-CS; if $R$ is a right max ring, then $M$ is a dual essentially Baer module if and only if $M$ is d-CS-Rickart and $M$ has the SSSP-d-CS; if $M$ be a projective module and $\mathcal{P}(M)=0$, then $M$ is a quasi-Baer module if and only if every fully invariant submodule of $M$ is essential in a fully invariant direct summand of $M$, if and only if the right annihilator in $M$ of every ideal of $S$ is essential in a fully invariant direct summand of $M$. We also give some characterizations of projective quasi-Baer modules. The presented results yield the known results related to Baer modules and dual Baer modules.
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