Direct sums of ADS* modules

Abstract

A module M is called ADS* if for every direct summand N of M and every supplement K of N in M, we have \(M=N\oplus K\). In this work, we study direct sums of ADS* modules. Many examples are provided to show that this notion is not inherited by direct sums. It is shown that if a module M has a decomposition \(M=A\oplus B\) which complements direct summands such that A and B are mutually projective, then M is ADS*. The class of rings R, for which all direct sums of ADS* R-modules are ADS*, is shown to be exactly that of the right V-rings. We characterize the class of right perfect rings R for which \(R\oplus S\) is ADS* for every simple R-module S as that of the semisimple rings.