Some graphs related to submodules of a module

TL;DR

This paper introduces and studies two new graphs based on submodules of an R-module, exploring their properties and relationships in the context of module theory.

Contribution

It defines the second submodule intersection graph and prime submodule sum graph for modules, providing new tools for analyzing module structure.

Findings

Characterization of SSI(M) and PSS(M) graphs

Relationships between graph properties and module properties

Potential applications in module classification

Abstract

Let R be a commutative ring with identity and M be an R-module. In this paper, we introduce and investigate the second submodule intersection graph SSI(M) of M with vertices are nonzero proper submodules of M and two distinct vertices N and K are adjacent if and only if is a second submodule of M. Also, we introduce and consider the prime submodule sum graph PSS(M) of M with vertices are non-zero proper submodules of M and two distinct vertices N and K are adjacent if and only if N + K is a prime submodule of M.