Closures of proper classes

Abstract

For an integral domain R we consider the closures M (Mr r ε R) of a submodule M of an R-module N consisting of elements n of N with tn 2 M (rmn ε M) for some nonzero t ε R (m ε Z+) and its connections with usual closure M of M in N. Using these closures we study the closures P and Pr of a proper class P of short exact sequences and give a decomposition for the class of quasi-splitting short exact sequences of abelian groups into the direct sum of "p-closures" of the class Split of splitting short exact sequences and description of closures of some classes. In the general case of an arbitrary ring we generalize these closures of a proper class P by means of homomorphism classes F and G and prove that under some conditions this closure is a propier classes. © 2020 Elsevier B.V. All rights reserved.