The Homomorphism Submodule Graph

TL;DR

This paper introduces the homomorphism submodule graph for modules, exploring its properties, computing it for specific classes, and showing it can determine module isomorphism over certain rings, revealing deep algebraic insights.

Contribution

It defines the homomorphism submodule graph, computes it for specific modules, and proves it determines module isomorphism over Artinian local rings.

Findings

Graph is chordal over commutative rings with identity.

Isomorphism type of modules over Artinian local rings is determined by the graph.

Spectral radius relates to module length in certain cases.

Abstract

Let $M$ be a left $R$-module. We define the \emph{homomorphism submodule graph} $\Gamma_{\mathrm{Hom}}(M)$ as the simple graph whose vertices are the proper submodules of $M$, with an edge between distinct vertices $N_1$ and $N_2$ if and only if $\mathrm{Hom}_R(N_1, M/N_2) \ne 0$ or $\mathrm{Hom}_R(N_2, M/N_1) \ne 0$. This graph encodes homological information about $M$ and reflects its internal structure. We compute $\Gamma_{\mathrm{Hom}}(M)$ for semisimple and uniserial modules, establish precise correspondences between graph-theoretic and algebraic properties, and prove that for modules over Artinian local rings, the isomorphism type of $M$ is determined by $\Gamma_{\mathrm{Hom}}(M)$. We also show that over commutative rings with identity, the graph is always chordal, and we relate its spectral radius to composition length in natural families.