The Schr\"{o}der-Bernstein problem for relative injective modules

TL;DR

This paper extends the Schr"{o}der-Bernstein theorem to the context of relative injective modules, establishing conditions under which two modules are isomorphic based on embeddings and injectivity properties.

Contribution

It provides a positive answer to a question about the isomorphism of modules under specific embedding and injectivity conditions within a relative framework.

Findings

Modules A and B are isomorphic under the given conditions.

The result generalizes classical Schr"{o}der-Bernstein theorem to module theory.

Conditions involve M-embeddings and K_M-injectivity.

Abstract

Let $(K,M)$ be a pair satisfying some mild condition, where $K$ is a class of $R$-modules and $M$ is a class of $R$-homomorphisms. We show that if $f:A\rightarrow B$ and $g:B\rightarrow A$ are $M$-embeddings and $A,B$ are $K_M$-injective, then $A$ is isomorphic to $B$, positively answering an question proposed by Marcos and Jiri [6].