Dual-ADS, ADS\(^\#\) and ADS* Modules

Abstract

A right R-module M is said to be dual-ADS if for every decomposition \(M=A\oplus B\) then A and B are mutually projective. The class of ADS*-modules contains the class of dual-ADS modules. In this article, we study several properties of these modules. It is shown that a module M is dual-ADS if and only if for any direct summand S and \(T^\prime \le M\) with \(T^\prime +S\) a direct summand of M, then \(T^\prime \) contains a direct complement of S in \(T^\prime +S\). A generalization of dual-ADS modules is considered, namely, ADS\(^\#\)-modules. It is shown that a module M is ADS\(^\#\) if and only if for any direct summand S of M, and any weak supplement \(T^\prime \) of S in \(T^\prime +S\) such that \(T^\prime +S\) is a direct summand of M, then \(T^\prime \) contains a direct complement of S in \(T^\prime +S\).