DIRECT SUMS OF MODULES HAVING (S∗)

Abstract

A module M is said to satisfy the property (S∗) if every submodule N of M is cosingular of a direct summand of M. In this study we investigate when a finite direct sum of modules with (S∗) satisfies (S ∗). We prove that a module M is a direct sum of modules satisfying (S∗) and Z ∗(M) has finite uniform dimension if and only if M = M1⊕M2⊕M3 where M1 is semisimple with Z∗(M1) = 0,M2 has finite uniform dimension with Z∗(M2)=M2 and M3 has finite uniform dimension and is a finite direct sum of local submodules of M.