An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms

作者：Macias Diaz Jorge E^*; Macias Siegfried

来源：Journal of Algebra and Its Applications, 2016, 15(9): 1650166.

DOI：10.1142/S0219498816501668

• Isomorphism of modules
• equivalence of modules
• generalized injective object
• algebraic class of morphisms
• generalized injective hull
• Bumby's theorem
• Cantor-Bernstein-Schroder's theorem

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-Schroder'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.

• 出版日期2016-11