Notes on Baer modules and their dual

Abstract

In this paper, we give new categorical characterizations of (dual-)Baer modules and then several applications of them are presented. Among other things, it is proved that a module $M_R$ is Baer if and only if for every $N\leq M_R$, $Rej^{-1}(N)$  is a direct summand of $M_R$. This shows that a module $M_R$ is Baer and co-retractable if and only if it is semisimple. Hence, over a ring Morita equivalent to a perfect duo ring, all Baer modules are  semisimple. If  $R$ is a right semi-hereditary and u.$\dim(R_R)$ is finite, then every finitely generated  torsionless $R$-module $M$ is Baer. Dually,  dual-Baer modules over certain rings are also investigated.  If  the  $R$-module $M^+ $ = $Hom_{\Bbb Z}(M,{\Bbb Q}/{\Bbb Z})$  is Baer, then it is shown that $Hom_R(M,N)M$ is a pure submodule $M_R$ for any $N\leq M_R$.