Zero-divisor graphs of reduced Rickart *-rings

Abstract

For a ring A with an involution *, the zero-divisor graph of A, Γ*(A), is the graph whose vertices are the nonzero left zero-divisors in A such that distinct vertices x and y are adjacent if and only if xy* = 0. In this paper, we study the zero-divisor graph of a Rickart *-ring having no nonzero nilpotent element. The distance, diameter, and cycles of Γ*(A) are characterized in terms of the collection of prime strict ideals of A. In fact, we prove that the clique number of Γ*(A) coincides with the cellularity of the hullkernel topological space Σ(A) of the set of prime strict ideals of A, where cellularity of the topological space is the smallest cardinal number m such that every family of pairwise disjoint non-empty open subsets of the space have cardinality at most m.

Keywords

• reduced ring
• rickart *-ring
• zero-divisor graph
• prime strict ideals
• primary: 16w10
• secondary: 05c25, 05c15