CLOSURES OF PROPER CLASSES


Alizade R., DEMİRCİ Y. M.

MISKOLC MATHEMATICAL NOTES, vol.17, no.2, pp.723-738, 2017 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 17 Issue: 2
  • Publication Date: 2017
  • Doi Number: 10.18514/mmn.2016.1566
  • Title of Journal : MISKOLC MATHEMATICAL NOTES
  • Page Numbers: pp.723-738
  • Keywords: proper class of short exact sequences, closure of a proper class, sum of proper classes, closure of a module

Abstract

For an integral domain R we consider the closures (M) over cap ((M) over cap (r), r epsilon R) of a submodule M of an R-module N consisting of elements n of N with t n epsilon M (r(m) n epsilon M) for some nonzero t epsilon R (m epsilon Z(+)) and its connections with usual closure (M) over bar of M in N. Using these closures we study the closures (P) over cap and (P) over cap (r) 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 (P) over cap (G)(F) is a proper classes.