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

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 property (P) if 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 ∈ Mod-R | Z_R(M) = 0} is closed under factor modules, then R has (P) if and only if the ring R is von Neumann regular.