Book Description

Abstract: We introduce the notions of the Baer and the 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. Every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the extending property and the FI-extending property are exhibited 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. While we show that direct sums of (quasi- ) Baer modules are not (quasi- ) Baer, we prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and that every free (projective) module over a quasi-Baer ring is always a quasi-Baer module. Some results, related to direct sums of Baer modules and direct sums of quasi-Baer modules, are also included. A ring over which every module is Baer is shown to be precisely a semisimple Artinian ring. 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. A characterization for this to hold in the Baer modules case is obtained. We provide a type theory of Baer modules and decomposition of a Baer module into into five types, similar to the one provided by Kaplansky for the Baer rings case. This type theory and type decomposition is applied, in particular, to all nonsingular extending modules. Applications of the results obtained are included.