Properties of \(P\)-coherent and Baer modules. (Q532118)

A left \(R\)-module \(M\) is a Baer module if the annihilator of any subset of \(M\) is a direct summand in \(R\) as left module. \(M\) is coherent if each finitely generated submodule of \(M\) is finitely presented. If this happens only for cyclic submodules, \(M\) is \(P\)-coherent (\(P\) from principal) and if each of its principal submodules is projective, then \(M\) is a PP-module. In the paper, some strong connections among these characteristics are established, as well as some nice properties of these classes of modules. The right \(R\)-modules considered as left \(S\)-modules, with \(S\) the ring of \(R\)-endomorphisms of \(M\), are investigated concerning the above properties. The left \(S\)-module \(M\) is called AFG if the \(M\)-annihilator of any subset of \(R\) is a finitely generated left \(S\)-module of \(M\). The connection between AFG-modules and \(P\)-coherent modules is studied.