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

Authors : Derya Keskin Tütüncü, Nil Orhan Ertaş, Patrick F. Smith, Rachid Tribak

Pages : 649-657

Doi:10.3906/mat-1210-15

View : 14 | Download : 11

Publication Date : 9999-12-31

Article Type : Makaleler

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