The Zariski topology-graph on the maximal spectrum of modules over commutative rings

TL;DR

This paper introduces a new graph called the Zariski topology-graph on the maximal spectrum of modules over commutative rings, linking algebraic and topological properties via graph theory.

Contribution

It defines the Zariski topology-graph on Max(M) and demonstrates its utility in analyzing module properties through graph-theoretic methods.

Findings

The graph provides insights into the structure of modules.

It connects topological properties with algebraic characteristics.

The approach offers new tools for module analysis.

Abstract

Let M be a module over a commutative ring and let Spec(M) (resp. Max(M)) be the collection of all prime (resp. maximal) submodules of M. We topologize Spec(M) with Zariski topology, which is analogous to that for Spec(R), and consider Max(M) as the induced subspace topology. For any non-empty subset T of Max(M), we introduce a new graph G(\tau^{m}_{T})called the Zariski topology-graph on the maximal spectrum of M. This graph helps us to study the algebraic (resp. topological) properties of M (resp. Max(M)) by using the graph theoretical tools.