ZERO-DIVISOR GRAPHS OF REDUCED RICKART ∗-RINGS
Journal Title: Discussiones Mathematicae - General Algebra and Applications - Year 2017, Vol 37, Issue 1
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.
Authors and Affiliations
A. A. Patil, B. N. Waphare
Keywords
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