On \tilde{Spec}(M) Topology of Module M over Commutative Rings

Abstract

Let R be a commutative ring with unity and M be an R-module. In this study, we construct the \tilde{Spec}(M) topology using the prime spectrum of module M and multiplicatively closed subsets of R with the closed sets \tilde{V}(S)={P \in Spec(M) : (P : M) \cap S_i \neq \emptyset for all i \in I} with the open sets \tilde{D}(S_i):={P \in Spec(M) : (P : M) \cap S_i = \emptyset} where S = {S_i}_{i \in I} is a family of multiplicatively closed subsets of R. We investigate connections between the algebraic properties of R-module M and the topological properties of \tilde{Spec}(M). We examine specifically the separation axioms, connectivity, nested and Lindel\"of property together with quasi-compactness as well as the isolated, closure, interior and limit points of tilde{Spec}(M). Moreover, in the last section, we provide an example of a Lindel\"of space which is not quasi-compact by means of \tilde{Spec}(M).