Physical Sciences and Mathematics Commons^™

Some Rings For Which The Cosingular Submodule Of Every Module Is A Direct Summand, Derya Keski̇n Tütüncü, Ni̇l Orhan Ertaş, Patrick F. Smith, Rachid Tribak

Turkish Journal of Mathematics

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 …