The number of minimal prime ideals in a k-integral domain and an algorithm for construction maximal $k$-zero divisors in $Z_n$

Abstract

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 $k\ge 2,$ 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 ${\mathbb{Z}}_n$.