Modules That Have a Weak Supplement in Every Extension

Abstract

We say that over an arbitrary ring a module M has the property (W E) (respectively, (W E E)) 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 (W E). A ring R is left perfect iff every left R-module has the property (W E) iff every left R-module has the property (W E E). 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 (W E*)(respectively, (W E E*)). © 2016 Miskolc University Press.