Orderings and preorderings on modules

Abstract

The purpose of this paper is to investigate some relevant properties of semireal modules. By introducing the notion of preorderings on modules over commutative rings, we discuss the interplay between semireality, preorderings and orderings on modules. In particular, we obtain that an R−module M is semireal if and only if M possesses a preordering. We also give some necessary and sufficient conditions for a preordering to be an ordering on modules.