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 $R$‎.