Prime ideal graphs of commutative rings

Abstract

<p style="text-align: justify;">Let <em>R</em> be a finite commutative ring with identity and <em>P</em> be a prime ideal of <em>R</em>. The vertex set is <em>R - </em>{0} and two distinct vertices are adjacent if their product in <em>P</em>. This graph is called the prime ideal graph of <em>R</em> and denoted by Γ<sub>P</sub>. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that Γ<sub>P</sub> 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<sub>P.</sub></p>

Topological Indices and Properties of the Prime Ideal Graph of a Commutative Ring and its Line Graph

“…Definition 2.1. [27] For a commutative ring R and its prime ideal P, the prime ideal graph P Γ is defined where the vertex set is R\{0}, and two vertices r 1 and r 2 are adjacent whenever…”

“…A graph representing a commutative ring focusing on prime ideals was introduced by Salih et al [27] in 2022. This graph is known as the prime ideal graph.…”

“…Two vertices, x and y are joined by an edge if and only if xy belongs to a prime ideal. The obtained results include chromatic, clique, independence, and domination numbers [27 graph's properties, such as its diameter being greater than or equal to 2 and its girth being greater than or equal to 4. In 2021, Asir et al [19] proposed one of the topological indices for ℤ n , the Wiener index, for its zero-divisor graph.…”

On Properties of Prime Ideal Graphs of Commutative Rings

“…This graph was later named as prime ideal graph. They defined a prime ideal graph 𝑃 of ring 𝑅, denoted by 𝛤 𝑃 , as a graph where the set of vertices is 𝑅 ∖ {0} and two vertices 𝑟 1 , 𝑟 2 are adjacent if and only if 𝑟 1 𝑟 2 ∈ 𝑃 [10]. Since the definition of prime ideal graph in that research is made to find relationships between prime ideal graphs and zero-divisor graphs, 0 is not included as a vertex in prime ideal graphs.…”