Classification of nonlocal rings with genus one 3-zero-divisor hypergraphs. Issue 1 (2nd January 2017)

• Record Type: Journal Article
• Authors: Selvakumar, K.; Ramanathan, V.
• Is Part Of: Communications in algebra. Volume 45:Issue 1(2017)
• Page Start: 275
• Page End: 284
• Publication Date: 2017-01-02
• Subjects: Hypergraph -- incidence graph -- planar hypergraph -- toroidal graph -- zero-divisor graph; 05C12 -- 05C25 -- 05C69 -- 13A99; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1206344
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let R be a commutative ring with identity and let Z ( R, k ) be the set of all k -zero-divisors in R and k >2 an integer. The k -zero-divisor hypergraph of R, denoted by ℋ k ( R ), is a hypergraph with vertex set Z ( R, k ), and for distinct elements x 1, x 2, …, x k in Z ( R, k ), the set { x 1, x 2, …, x k } is an edge of ℋ k ( R ) if and only if ∏ i = 1 k x i = 0 and the product of any ( k −1) elements of { x 1, x 2, …, x k } is nonzero. In this paper, we characterize all finite commutative nonlocal rings R with identity whose ℋ 3 ( R ) has genus one.