Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

• 作者: K.Selvakumar M.Subajini
• 来源期刊: 代数集刊（英文版）
• 年，卷（期）: 2021,（4）
• 页码范围: 655-672
• 页数: 18页
• 语种: 英文
• DOI: 10.1142/S1005386721000511

摘要

Let R be a commutative ring,I an ideal of R and k ≥ 2 a fixed integer.The ideal-based k-zero-divisor hypergraph HkI(R) of R has vertex set ZI(R,k),the set of all ideal-based k-zero-divisors of R,and for distinct elements x1,x2,…,xk in ZI(R,k),the set {x1,x2,…,xk} is an edge in HkI(R) if and only if x1x2…xk ∈ I and the product of the elements of any (k-1)-subset of {x1,x2,…,xk} is not in I.In this paper,we show that H3I(R) is connected with diameter at most 4 provided that x2 (∈) I for all ideal-based 3-zero-divisor hypergraphs.Moreover,we find the chromatic number of H3 (R) when R is a product of finite fields.Finally,we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H3I (R) planar.