Modules That Have A Weak Supplement in Every Extension

Abstract

We say that over an arbitrary ring a module M has the property. (WE) (respectively, (WEE)) if M has a weak supplement (respectively, ample weak supplements) in every extension. In this paper, we provide various properties of modules with these properties. We show that a module M has the property. (WEE) iff every submodule of M has the property. (WE). A ring R is left perfect iff every left R-module has the property. (WE) iff every left R-module has the property. (WEE). A ring R is semilocal iff every left R-module has a weak supplement in every extension with small radical. We also study modules that have a weak supplement(respectively, ample weak supplements) in every coatomic extension, namely the property. (WE*) (respectively,. (WEE*)).