s-WH Modules

Abstract

This article introduces the concept of strongly-WH module which is a proper generalized of Hopfian modules.  A module  is called strongly-WH, briefly -WH if, any -small surjective -endomorphism of  is an automorphism. We specify and provide some properties of this concept. Furthermore, we have established connections between strongly-WH modules and various other concepts. We demonstrate that every strongly-WH module is -weakly Hopfian. As well as, we provide cases in which the concepts of WH, -weakly Hopfian, and strongly-WH modules are equivalent.