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Let \(R\) be a commutative ring with identity. For any integer \(K > 1,\) an element is a \(k\) zero divisor if there are \(K\) distinct elements including the given one, such that the product of all is zero but the product of fewer than all is nonzero. Let \(Z(R,K)\) denote the set of the \(K\) zero divisors of \(R\). A ring with no \(K\)-zero divisors is called a \(K\)-domain. In this paper we define the hyper-graphic constant \(HG(R)\) and study some basic properties of \(K\)-domains. Our main results is theorem 5.1 which is as fellow:
Let \(R\) be a commutative ring such that the total ring of fraction \(T(R)\) is dimensional. If \(R\) is a \(K\)-domain for \(k \geq 2,\) then \(R\) has finitely many minimal prime ideals.
Using the results and lemma 5.4, we improve a finiteness theorem proved in [11] to a more robust theorem 5.5 which says:
Suppose \(R\) is not a \(k\)-domain and has more then \(k\)-minimal prime ideals.
Further, suppose that \(T(R)\) is a zero dimensional ring. Then \(Z(R,K)\) is finite if and only if \(R\) is finite.
We end this paper with a proof of an algorithm describing the maximal \(k\)-zero divisor hypergraphs on \(\mathbb{Z}_n\).