Isomorphisms between injective modules

TL;DR

This paper generalizes classical isomorphism results for modules within certain injective structures, showing that mutual embeddability implies isomorphism under specific conditions, extending known cases like pure and RD-injective modules.

Contribution

It introduces a broad framework for module isomorphisms in injective structures, generalizing previous specific results and encompassing various classes of injective modules.

Findings

Modules in the class are isomorphic if there are mutual maps in the structure.

Includes examples like pure, coneat, and RD-injective modules.

Generalizes classical isomorphism theorems for 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.