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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2511.07837 |
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| _version_ | 1866912701336256512 |
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| author | Mehry, Shahram Molaeinejad, Mansour |
| author_facet | Mehry, Shahram Molaeinejad, Mansour |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07837 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Homomorphism Submodule Graph Mehry, Shahram Molaeinejad, Mansour Combinatorics Commutative Algebra 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. |
| title | The Homomorphism Submodule Graph |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2511.07837 |