Baer and quasi-Baer modules

We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) K-cononsingular if and only if it is (FI-) extending and (FI-) K--nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.