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Title:
Extended annihilating-ideal graph of a commutative ring
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Discussiones Mathematicae - General Algebra and Applications 42(2) (2022) 279-291
Received: 2020-06-27 , Revised: 2021-02-04 , Accepted: 2022-03-22 , Available online: 2022-10-05 , https://doi.org/10.7151/dmgaa.1390
Abstract:
Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$ is
called an annihilating-ideal if there exists a nonzero ideal $J$ of $R$ such
that $IJ = (0)$ and we use the notation $\mathbb A(R)$ for the set of all
annihilating-ideals of $R$. In this paper, we introduce the extended
annihilating-ideal graph of $R$, denoted by $\mathbb E\mathbb A\mathbb G(R)$.
It is the simple graph with vertices $\mathbb A(R)^* =\mathbb A(R)\backslash
\left\{(0)\right\}$, and two distinct vertices $I$ and $J$ are adjacent whenever
there exist two positive integers $n$ and $m$ such that $I^nJ^m = (0)$ with
$I^n \neq (0)$ and $J^m \neq (0)$. Here we discuss in detail the diameter and
girth of $\mathbb E\mathbb A\mathbb G(R)$ and investigate the coincidence of
$\mathbb E\mathbb A\mathbb G(R)$ with the annihilating-ideal graph
$\mathbb A\mathbb G(R)$. Moreover we propose open questions in this paper.
Keywords:
annihilating-ideal graph, extended annihilating-ideal graph
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