# The Schr\"{o}der-Bernstein problem for Modules

## TL;DR

This paper investigates the Schröder-Bernstein problem within module theory, providing positive solutions for modules invariant under certain envelope endomorphisms, including injective and pure-injective cases.

## Contribution

It extends the Schröder-Bernstein problem to modules invariant under endomorphisms of their envelopes, especially for injective and pure-injective modules, under mild conditions.

## Key findings

- Positive solution for modules invariant under envelope endomorphisms.
- Extension of results to modules invariant under automorphisms of envelopes.
- Applicable to injective, pure-injective, and cotorsion envelopes.

## Abstract

In this paper we study the Schr\"{o}der-Bernstein problem for modules. We obtain a positive solution for the Schr\"{o}der-Bernstein problem for modules invariant under endomorphisms of their general envelopes under some mild conditions that are always satisfied, for example, in the case of injective, pure-injective or cotorsion envelopes. In the particular cases of injective envelopes and pure-injective envelopes, we are able to extend it further and we show that the Schr\"{o}der-Bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure-injective envelopes.