The Zariski Topology-Graph of Modules Over Commutative Rings

Abstract

Let M be a module over a commutative ring, and let Spec(M) be the collection of all prime submodules of M. We topologize Spec(M) with Zariski topology, which is analogous to that for Spec(R), and for a nonempty subset T of Spec(M), we introduce a new graph G(τ _T ), called the Zariski topology-graph. This graph helps us to study the algebraic (resp. topological) properties of M (resp. Spec(M)) by using the graph theoretical tools.

Key Words:

• Annihilating-submodule
• Graph
• Prime submodules
• Vertices
• Zariski topology

2010 Mathematics Subject Classification:

• 13C13
• 13C99
• 05C75

ACKNOWLEDGMENTS

We would like to thank the referee for comments and the careful reading of our manuscript.

Notes

Communicated by I. Swanson.