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Some graphs related to submodules of a module

Faranak Farshadifar

TL;DR

This work extends graph-theoretic analyses from rings to modules by defining the second submodule intersection graph $SSI(M)$ and the prime submodule sum graph $PSS(M)$ on non-zero proper submodules of an $R$-module $M$, with adjacencies given by $N \cap K$ being a second submodule or $N + K$ being a prime submodule, respectively. It analyzes structural properties such as universal vertices, connectivity, diameter, and domination under module-specific hypotheses, and connects these graphs to their ring-theoretic counterparts $SII(R)$ and $PIS(R)$, particularly in multiplication and hollow module settings. Key contributions include precise universal-vertex characterizations, conditions for completeness and connectedness, and reductions to annihilator graphs in multiplication modules, thereby translating ring-graph phenomena into the module context. The results provide new module-theoretic graph invariants and illuminate how submodule lattice properties influence graphical structure and potential applications in algebraic graph theory.

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.

Some graphs related to submodules of a module

TL;DR

This work extends graph-theoretic analyses from rings to modules by defining the second submodule intersection graph and the prime submodule sum graph on non-zero proper submodules of an -module , with adjacencies given by being a second submodule or being a prime submodule, respectively. It analyzes structural properties such as universal vertices, connectivity, diameter, and domination under module-specific hypotheses, and connects these graphs to their ring-theoretic counterparts and , particularly in multiplication and hollow module settings. Key contributions include precise universal-vertex characterizations, conditions for completeness and connectedness, and reductions to annihilator graphs in multiplication modules, thereby translating ring-graph phenomena into the module context. The results provide new module-theoretic graph invariants and illuminate how submodule lattice properties influence graphical structure and potential applications in algebraic graph theory.

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.
Paper Structure (3 sections, 31 theorems, 3 equations)

This paper contains 3 sections, 31 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Second submodule intersection graph of a module
  3. Prime submodule sum graph of a module

Key Result

Theorem 2.2

Let $M$ be an $R$-module such that every submodule contains a minimal submodule. Then $SSI(M)$ has a universal vertex if and only if one of the two statements hold:

Theorems & Definitions (66)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 56 more