On the genus of the kk-maximal hypergraph of commutative rings
Abstract
Let RR be a commutative ring with identity and k≥2k≥2, a fixed integer. Let ℐ(R,k) be the set of all k-maximal elements in R. Associate a k-maximal hypergraphℋk(R) to R with vertex set ℐ(R,k) and for distinct elements a1,a2,…,ak in ℐ(R,k), the set {a1,a2,…,ak} is an edge of ℋk(R) if and only if ∑ki=1Rai=R and ∑ki=1i≠jRai≠R for all 1≤j≤k. In this paper, we determine all isomorphism classes of finite commutative non-local rings with identity whose k-maximal hypergraph has genus one. Finally, we classify all finite commutative non-local rings R for which ℋ3(R) is projective.
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