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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  • EP ID EP394547
  • DOI 10.7151/dmgaa.1265
  • Views 53
  • Downloads 0

How To Cite

A. A. Patil, B. N. Waphare (2017). Zero-divisor graphs of reduced Rickart *-rings. Discussiones Mathematicae - General Algebra and Applications, 37(1), 31-43. https://europub.co.uk/articles/-A-394547