Isomorphisms between injective modules

Abstract

Suppose that $(\mathcal{F},\mathcal{M})$ is an injective structure of $R$-Mod such that the class $\mathcal{F}$ is closed for direct limits, then two modules in $\mathcal{M}$ are isomorphic if there are maps in $\mathcal{F}$ from each one of the modules into the other. Examples of module classes in such injective structures include (pure, coneat, and RD-) injective modules, as well as $\tau$-injective modules for a hereditary torsion theory $\tau$. Thus providing a generalization of a classical result of Bumby's and two recent ones by Mac\'{i}as-D\'{i}az.