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

Abstract

The submodule Z(M ) = ∩{N | M/N is small in its injective hull } was introduced by Talebi and Vanaja in 2002. A ring R is said to have propertyshown 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 ∈ M od − R | ZR(M ) = 0} is closed under factor modules, then R has ( P ) if and only if the ring R is von Neumann regular.