Behavior of countably generated pure-projective modules
A short exact sequence \(0 \to A \to B \to C \to 0\) of left \(R\)-modules is said to be pure, if for any right \(R\)-module \(M\), the canonical mapping \(0 \to M \otimes_ R A \to M \otimes_ R B\) is an injection. Under these conditions, \(A\) is said to be a pure submodule of \(B\). A left \(R\)- module \(M\) is called pure-projective if \(\text{Hom}_ R(M,B) \to \text{Hom}_ R(M,C) \to 0\) is exact for every pure short exact sequence \(0 \to A \to B \to C \to 0\). Pure-injectives are defined in a similar fashion. The author calls a ring \(R\) left pure-semisimple if it satisfies the equivalent conditions: 1) every left \(R\)-module is pure-injective, and 2) every left \(R\)-module is pure-projective. Left pure-semisimple rings are known to be left Artinian and they have been characterized as those rings which satisfy the following equivalent conditions: 1) every left \(R\)-module is a direct sum of finitely generated submodules, 2) every left \(R\)-module is a direct sum of indecomposable submodules, and 3) every indecomposable left \(R\)-module is finitely presented. The author adds to these results by giving additional characterizations of left pure-semisimple rings in terms of countably generated pure-projective modules. He shows that a ring \(R\) is left pure-semisimple if and only if every countably generated pure-projective left \(R\)-module is pure- injective. This result shows that \(R\) is left pure-semisimple if and only if every countably presented left \(R\)-module is pure-injective. Finally, he shows that \(R\) is left pure-semisimple if and only if for any countably generated pure-projective left \(R\)-module \(P\), every countably generated pure-projective pure submodule of \(P\) is a direct summand of \(P\).
countably generated pure-projective modules
scientific article; zbMATH DE number 203305