\(k\)-Zero Divisor Hypergraphs on Commutative Rings

Let \(R\) be a commutative ring with identity. For any integer \(k > 1\), an element is a \(k\)-zero divisor if there are distinct \(k\) 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 \(R\). In this paper we consider rings which are not \(k\)-integral domains (i.e. \(Z(R,k)\) is nontrivial) with finite \(Z(R,k)\). We show that a uniform \(n\) exists such that \(a^n = 0\) for all elements \(a\) of the nil-radical \(N\) and deduce that a ring \(R\) which is not a \(k\)-integral domain with more than \(k\) minimal prime ideals and whose nil-radical is finitely generated is finite, if \(Z(R,k)\) is finite.