A generalization of the annihilating ideal graph for modules

Soraya Barzegar ^[1] ; Saeed Safaeeyan ^[1] ; Ehsan Momtahan ^[1]
1. [1] Yasouj University
  
  Yasouj University
  
  Irán
Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 65, Nº. 1, 2023, págs. 47-65
Idioma: inglés
DOI: 10.33044/revuma.2158
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Resumen
- We show that an R-module M is noetherian (resp., artinian) if and only if its annihilating submodule graph, G(M), is a non-empty graph and it has ascending chain condition (resp., descending chain condition) on vertices. Moreover, we show that if 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 Γ(Zn) and G(Zn) by Maple, simultaneously.