The Homomorphism Submodule Graph

TL;DR

The paper introduces the homomorphism submodule graph $\Gamma_{\mathrm{Hom}}(M)$ to encode homological relations among submodules of $M$ via nonzero $R$-homomorphisms to quotients. It provides detailed structural results for semisimple and uniserial modules, proves a reconstruction theorem showing that $M$ is determined by $\Gamma_{\mathrm{Hom}}(M)$ for Artinian local rings, and establishes that over commutative rings the graph is chordal with diameter at most $2$. It further links spectral data to composition length in natural families, and outlines fertile directions including broader chordality characterizations, Laplacian spectral connections to depth and projective dimension, and a derived analogue. These results collectively connect homological module theory with graph-theoretic and spectral invariants, offering a combinatorial handle on module isomorphism questions and new avenues for invariants.

Abstract

Let $M$ be a left $R$-module. We define the \emph{homomorphism submodule graph} $Γ_{\mathrm{Hom}}(M)$ 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 $Γ_{\mathrm{Hom}}(M)$ 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 $Γ_{\mathrm{Hom}}(M)$. 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.

Theorems & Definitions (39)

• Definition 1
• Example 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: Complete Graph
• proof
• Theorem 4: Connectedness
• proof