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

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 x^(2)(∈)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.