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

Authors

DERYA KESKİN TÜTÜNCÜ
NİL ORHAN ERTAŞ
PATRICK F. SMITH
RACHID TRIBAK

DOI

10.3906/mat-1210-15

Abstract

The submodule \overline{Z}(M) = \cap {N 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 \overline{Z}(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 \in Mod-R \overline{Z}_{R}(M) = 0} is closed under factor modules, then R has (P) if and only if the ring R is von Neumann regular.

Keywords

von Neumann regular ring, perfect ring, (non)cosingular submodule

First Page

649

Last Page

657

Recommended Citation

TÜTÜNCÜ, DERYA KESKİN; ERTAŞ, NİL ORHAN; SMITH, PATRICK F.; and TRIBAK, RACHID (2014) "Some rings for which the cosingular submodule of every module is a direct summand," Turkish Journal of Mathematics: Vol. 38: No. 4, Article 4. https://doi.org/10.3906/mat-1210-15
Available at: https://journals.tubitak.gov.tr/math/vol38/iss4/4