On the genus of the k-annihilating-ideal hypergraph of commutative rings

K. Selvakumar () and V. Ramanathan
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K. Selvakumar: Manonmaniam Sundaranar University
V. Ramanathan: Manonmaniam Sundaranar University

Indian Journal of Pure and Applied Mathematics, 2019, vol. 50, issue 2, 461-475

Abstract: Abstract Let R be a commutative ring and k an integer greater than 2 and let $${\cal A}(R, k)$$ A ( R , k ) be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by $${\cal A}{{\cal G}_k}(R)$$ A G k ( R ) , is a hypergraph with vertex set $${\cal A}(R, k)$$ A ( R , k ) , and for distinct elements I1, …, Ik in $${\cal A}(R, k)$$ A ( R , k ) , the set {I1, I2, …, Ik} is an edge of $${\cal A}{{\cal G}_k}(R)$$ A G k ( R ) if and only if $$\prod\limits_{i = 1}^k {{I_i} = (0)} $$ ∏ i = 1 k I i = ( 0 ) and the product of any (k − 1) elements of the set {I1, I2,…, Ik} is nonzero. In this paper, we characterize all Artinian commutative nonlocal rings R whose $${\cal A}{{\cal G}_3}(R)$$ A G 3 ( R ) has genus one.

Keywords: k-zero-divisor hypergraph; k-annihilating-ideal hypergraph; incidence graph; toroidal graph (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s13226-019-0338-3

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