$k$-Zero Divisor Hypergraphs on Commutative Rings

Abstract

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.