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 \(\mathcal {H}_k(R)\), is a hypergraph with vertex set Z(R, k), and for distinct elements \(x_1,x_2,\ldots ,x_k\) in Z(R, k), the set \(\{x_1,x_2, \ldots , x_k\}\) is an edge of \(\mathcal {H}_k(R)\) if and only if \(\prod \limits _{i=1}^kx_i =0\) and the product of any \((k-1)\) elements of \(\{x_1,x_2,\ldots ,x_k\}\) is nonzero. In this paper, we characterize all finite commutative non-local rings R with identity whose \(\mathcal {H}_3(R)\) has crosscap one.
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The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions. The work reported here is supported by the UGC-Major Research Project [F.No. 42-8/2013(SR)] awarded to K. Selvakumar by the University Grants Commission, Government of India.
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Selvakumar, K., Ramanathan, V. Classification of non-local rings with projective 3-zero-divisor hypergraph. Ricerche mat 66, 457–468 (2017). https://doi.org/10.1007/s11587-016-0313-9
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DOI: https://doi.org/10.1007/s11587-016-0313-9