On (co)pure Baer injective modules

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.