On (co)pure Baer injective modules

Abstract

For a given class of \(R\)-modules \(\mathcal{Q}\), a module \(M\) is called \(\mathcal{Q}\)-copure Baer injective if any map from a \(\mathcal{Q}\)-copure left ideal of \(R\) into \(M\) can be extended to a map from \(R\) into \(M\). Depending on the class \(\mathcal{Q}\), this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as \(\mathcal{Q}\)-copure submodule of a \(\mathcal{Q}\)-copure Baer injective module. Certain types of rings are characterized using properties of \(\mathcal{Q}\)-copure Baer injective modules. For example a ring \(R\) is \(\mathcal{Q}\)-coregular if and only if every \(\mathcal{Q}\)-copure Baer injective \(R\)-module is injective.