Revista de la
Unión Matemática Argentina
A generalization of the annihilating ideal graph for modules
Soraya Barzegar, Saeed Safaeeyan, and Ehsan Momtahan
Volume 65, no. 1 (2023), pp. 47–65    

https://doi.org/10.33044/revuma.2158

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Abstract

We show that an $R$-module $M$ is noetherian (resp., artinian) if and only if its annihilating submodule graph, $\mathbb{G}(M)$, is a non-empty graph and it has ascending chain condition (resp., descending chain condition) on vertices. Moreover, we show that if $\mathbb{G}(M)$ is a locally finite graph, then $M$ is a module of finite length with finitely many maximal submodules. We also derive necessary and sufficient conditions for the annihilating submodule graph of a reduced module to be bipartite (resp., complete bipartite). Finally, we present an algorithm for deriving both $\Gamma (\mathbb{Z}_n)$ and $\mathbb{G}(\mathbb{Z}_n)$ by Maple, simultaneously.

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