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

TL;DR

This paper introduces a new topology Spec(M) on modules over commutative rings, exploring its topological properties and their relation to algebraic features of the module.

Contribution

It constructs the Spec(M) topology using prime spectra and multiplicative subsets, and analyzes its topological properties in relation to module algebraic properties.

Findings

Spec(M) topology satisfies various separation axioms

The topology's properties relate to module algebraic features

An example shows Lindelfof space not necessarily quasi-compact

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).