Journal of Algebra and Its Applications Vol. 15, No. 9 (2016) 1650166 (13 pages) c  World Scientific Publishing Company DOI: 10.1142/S0219498816501668 An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms Jorge E. Mac´ ıas-D´ ıaz Departamento de Matem´ aticas y F´ ısica Universidad Aut´ onoma de Aguascalientes Avenida Universidad 940, Ciudad Universitaria Aguascalientes, Aguascalientes 20131, Mexico jemacias@correo.uaa.mx Siegfried Mac´ ıas Centro de Ciencias B´ asicas Universidad Aut´ onoma de Aguascalientes Avenida Universidad 940, Ciudad Universitaria Aguascalientes, Aguascalientes 20131, Mexico sieg macias@hotmail.com Received 13 February 2015 Accepted 6 October 2015 Published 19 November 2015 Communicated by J. Trlifaj Dedicated to Professor L´aszl´o Fuchs on the occasion of his 90th birthday. Departing from a general definition of injectivity of modules with respect to suitable algebraic classes of morphisms, we establish conditions under which two modules are isomorphic when they are isomorphic to submodules of each other. The main result of this work extends both Bumby’s criterion for the isomorphism of injective modules and the well-known Cantor–Bernstein–Schr¨oder’s theorem on the cardinality of sets. In the way, various properties on essential extensions, injective modules and injective hulls are generalized. The applicability of our main theorem embraces the cases of RD-injective and pure-injective modules as particular scenarios. Many of the propositions which lead to the proof of the main result of this paper are valid for arbitrary categories. Keywords : Isomorphism of modules; equivalence of modules; generalized injective object; algebraic class of morphisms; generalized injective hull; Bumby’s theorem; Cantor– Bernstein–Schr¨oder’stheorem. Mathematics Subject Classification: 13C10, 13C05, 13F05, 16D40 1650166-1