<p>The exploration of graphs linked to algebraic structures has emerged as a dynamic field, bridging graph theory and module theory to reveal deeper insights into module properties. In this study, we introduce and analyze the semisimple intersection graph, denoted as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$GSS_R(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <msub> <mi>S</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> associated with a right<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R $</EquationSource> <EquationSource Format="MATHML"><math> <mi>R</mi> </math></EquationSource> </InlineEquation>-module <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation>. The vertices of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$GSS_R(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <msub> <mi>S</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are the nonzero submodules of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation>, with two vertices <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$N$</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$K$</EquationSource> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> </InlineEquation> connected if their intersection forms a nonzero semisimple module. We examine fundamental properties of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$GSS_R(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <msub> <mi>S</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> including its connectivity, diameter, and domination number, in relation to the module-theoretic characteristics of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation>. Additionally, the girth of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$GSS_R(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <msub> <mi>S</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is determined.</p>

A specific intersection graph of a module applying semisimple property

期刊论文
发表日期：

2025年10月13日

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R. Ahmadzadeh,
A. R. Moniri Hamzekolaee

摘要

The exploration of graphs linked to algebraic structures has emerged as a dynamic field, bridging graph theory and module theory to reveal deeper insights into module properties. In this study, we introduce and analyze the semisimple intersection graph, denoted as $GSS_R(M)$ $G S S_{R} (M)$ associated with a right $R $ $R$ -module $M$ $M$ . The vertices of $GSS_R(M)$ $G S S_{R} (M)$ are the nonzero submodules of $M$ $M$ , with two vertices $N$ $N$ and $K$ $K$ connected if their intersection forms a nonzero semisimple module. We examine fundamental properties of $GSS_R(M)$ $G S S_{R} (M)$ including its connectivity, diameter, and domination number, in relation to the module-theoretic characteristics of $M$ $M$ . Additionally, the girth of $GSS_R(M)$ $G S S_{R} (M)$ is determined.