An extension of annihilating-ideal graph of commutative rings

Commun. Korean Math. Soc. 2020 Vol. 35, No. 4, 1045-1056 https://doi.org/10.4134/CKMS.c200006 Published online September 7, 2020 Printed October 31, 2020

Mahtab Koohi Kerahroodi, Fatemeh Nabaei P.O.Box: 65719-95863; Islamic Azad University

Abstract : Let $R$ be a commutative ring with unity. The extension of annihilating-ideal graph of $R$, $\overline{\mathbb{AG}}(R)$, is the graph whose vertices are nonzero annihilating ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if there exist $n,m\in\mathbb{N}$ such that $I^{n}J^{m}=(0)$ with $I^{n}, J^{m}\neq (0)$. First, we differentiate when $\mathbb{AG}(R)$ and $\overline{\mathbb{AG}}(R)$ coincide. Then, we have characterized the diameter and the girth of $\overline{\mathbb{AG}}(R)$ when $R$ is a finite direct products of rings. Moreover, we show that $\overline{\mathbb{AG}}(R)$ contains a cycle, if $\overline{\mathbb{AG}}(R)\neq\mathbb{AG}(R)$.