Characterization of χ-topological modules

Abstract

In this paper, we attempt to study the topological notions of multiplication module in the realm of purity. We let χ denote a collection of multiplication ideals of a ring R. We call an R module M to be χ-topological module (the topology being the Zariski topology on the prime spectrum of M) if, together with all the axiom of a topological space, it satisfies an additional condition that the collection of all the open sets is closed under arbitrary union. It is seen that over a Noetherian ring, a finitely generated multiplication module is a topological module if and only if it is a χ-topological module. Some important topological properties of a pure submodule of a multiplication module have also been studied.