THE EXTENDING CONDITION RELATIVE TO SETS OF SUBMODULES


Birkenmeier G. F. , TERCAN A. , CELEP YÜCEL C.

COMMUNICATIONS IN ALGEBRA, vol.42, no.2, pp.764-778, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 2
  • Publication Date: 2014
  • Doi Number: 10.1080/00927872.2012.723084
  • Title of Journal : COMMUNICATIONS IN ALGEBRA
  • Page Numbers: pp.764-778

Abstract

A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module ?-extending if every member of the set ? is essential in a direct summand where ? is a subset of the set of all submodules of M. Our focus is the behavior of the ?-extending modules with respect to direct sums and direct summands. By obtaining various well-known results on extending modules and generalizations as corollaries of our results, we show that the ?-extending concept provides a unifying framework for many generalizations of the extending notion. Moreover, by applying our results to various sets ?, including the projection invariant submodules, the projective submodules, and torsion or torsion-free submodules of a module, we obtain new results including a characterization of the projection invariant extending Abelian groups.