The Schröder–Bernstein problem for modules

Abstract

In this paper we study the Schröder–Bernstein problem for modules. We obtain a positive solution for the Schröder–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 Schröder–Bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure-injective envelopes.