The radical-depended graph of a commutative ring
Abstract
Let $R$ be a commutative ring with identity and $\sqrt{I}$ be the radical of an ideal $I$ of $R$. We introduce the radical-depended graph $\mathcal{G}_I(R)$ whose vertex set is $\{x\in R\setminus \sqrt{I}\mid xy\in I$ for some $y\in R\setminus \sqrt{I}\}$ and distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. In this paper, several properties of $\mathcal{G}_I(R)$ are investigated and some results on the parameters of this graph are given. It follows that if $I$ is a quasi primary ideal, then $\mathcal{G}_I(R)=\emptyset$. It is shown that if $I$ is a $2$-absorbing ideal of $R$ which is not quasi primary, then $\mathcal{G}_I(R)$ is the complete bipartite graph $K_{1,1}$ or $K_{m,n}$ for some $m,n\geq 2$. Moreover, it is proved that $\mathcal{G}_I(R)$ is a connected graph with diameter at most $3$, and if it has a cycle, then its girth is at most $4$. Also, it is shown that if $R$ is a Noetherian ring, then the clique number of $\mathcal{G}_I(R)$ is equal to $|\operatorname{Min}(R/I)|$ for every ideal $I$ of $R$.
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PDFDOI: https://doi.org/10.52846/ami.v45i1.896