Serre conditions for category of gradual modules

Abstract

In this paper, we first describe the gradual modules and the homomorphisms between two gradual modules. Next, we introduce the category G-Mod, whose objects are all gradual modules and morphisms are all homomorphisms between two gradual modules. Also, according to the definition of the gradual module’s structure induced on each homomorphism f, we consider the gradual module’s structure on ker(f) and Coker(f). We show that being a monomorphism (respectively, epimorphism) in the category G-Mod is equivalent to a monomorphism (respectively, epimorphism) in the category R-Mod. As the main result, we prove that in any short exact sequence 0→M'→M→M′′→0, in R-Mod, if the R-module M has a gradual module’s structure, then there are gradual module’s structure on the modules M' and M′′ where the short exact sequence created is a short exact sequence in the category G-Mod.