### Prime ideal graphs of commutative rings

#### Abstract

Let *R* be a finite commutative ring with identity and *P* be a prime ideal of *R*. The vertex set is *R - *{0} and two distinct vertices are adjacent if their product in *P*. This graph is called the prime ideal graph of *R* and denoted by Γ_{P}. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that Γ_{P} is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of Γ_{P.}

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PDFDOI: http://dx.doi.org/10.19184/ijc.2022.6.1.2

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