When mutually epimorphic modules are isomorphic

Abstract

The Schröder-Bernstein Theorem for sets is well-known. A dual of the Schröder-Bernstein Theorem is that two sets with surjection maps onto each other are isomorphic. Analogous to this dual, the question of whether two algebraic structures which are epimorphic to each other, are always isomorphic to each other, is of interest. For modules over a given ring, this does not hold true in general. For a ring R, a subclass C of R-modules is said to satisfy the dual of Schröder-Bernstein (or DSB) property if any pair of its members are isomorphic whenever each one is epimorphic image of the other. We investigate the DSB property for the classes of (quasi-)discrete and (quasi-)projective modules among other results in this paper. In particular, we prove that the class of discrete R-modules has the DSB property while the class of quasi-discrete modules does not satisfy the DSB property. On the other hand, over Dedekind domains and generalized uniserial rings, the DSB property holds for the class of quasi-discrete modules. We show that over a right (semi-)perfect ring R, the class of (finitely generated) quasi-projective R-modules satisfies the DSB property, however, over a (von-Neumman) regular ring R these classes do not satisfy the property. We also investigate the DSB property for the class of injective modules. As applications, our investigations provide answers to several open questions posed earlier in this regard.