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

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