Pseudo-Valuation Modules

Abstract

The aim of this paper is to generalize the notion of pseudo-valuation to modules over arbitrary commutative rings. We generalize the notion of strongly prime ideal, as defined in Badawi et al. (Lecture Notes in Pure and Applied Mathematics 185:57–67, 1997), to the notion of strongly prime submodule. We define a module M to be a pseudovaluation module if every prime submodule of M is strongly prime. It is shown that if M has a maximal submodule N, thenM is pseudo-valuation if and only if N is strongly prime. Also, we characterize strongly prime submodules in pseudo-valuation modules. We investigate some properties of these modules, and study relations between some structures and these modules.