A Note on Minimal Prime Divisors of an Ideal

Abstract

We explore several equivalent conditions for finiteness of the set of minimal prime divisors of an ideal and conclude the results of Anderson, Gilmer and Heinzer as especial cases. It is proved that in the ring of polynomials K[X], in which K is a Noetherian ring and X a (possibly infinite) set of indeterminates over K, these conditions are necessary and sufficient. In particular, it is proved that in K[X], an ideal has a finite number of minimal prime divisors if and only if all its minimal prime divisors have finite height. The same results are proved for the derived normal ring of a Noetherian integral domain and the quotient ring K[X]/I, in which I is generated by a K[X]-regular sequence of finite length. We also give a counterexample to show that the conditions of Anderson, Gilmer and Heinzer are not sufficient.