Some rings for which the cosingular submodule of every module is a direct summand


KESKİN TÜTÜNCÜ D. , ORHAN ERTAŞ N., Smith P. F. , TRIBAK R.

TURKISH JOURNAL OF MATHEMATICS, vol.38, no.4, pp.649-657, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 4
  • Publication Date: 2014
  • Doi Number: 10.3906/mat-1210-15
  • Title of Journal : TURKISH JOURNAL OF MATHEMATICS
  • Page Numbers: pp.649-657

Abstract

The snbmodule (Z)overbar(M) = boolean AND{N vertical bar M/N is small in its injective hull} was introduced by Talebi and Vanaja in 2002. A ring R is said to have property (P) if (Z)overbar(M) is a direct summand of M for every R-module M. It is shown that a commutative perfect ring R has (P) if and only if R is semisimple. An example is given to show that this characterization is not true for noncommutative rings. We prove that if R is a commutative ring such that the class {M is an element of Mod-R vertical bar <(Z)overbar >(R)(M) = 0} is closed under factor modules, then R has (P) if and only if the ring R is von Neumann regular.