Commutative rings whose prime ideals are radically perfect

Abstract

The main objective of this paper is to relate the height and the number of generators of ideals in rings that are not necessarily Noetherian. As in [{\bf10, 11}], we call an ideal $I$ of a ring $R$ radically perfect if among the ideals of $R$ whose radical is equal to the radical of $I$ the one with the least number of generators has this number of generators equal to the height of $I$. This is a generalization of the notion of set theoretic complete intersection of ideals in Noetherian rings to rings that need not be Noetherian. In this work, we determine conditions on a ring $R$ so that the prime ideals of $R$ and also those of the polynomial rings $R[X]$ over $R$ are radically perfect. In many cases, it is shown that the condition of prime ideals of $R$ or that of $R[X]$ being radically perfect is equivalent to a form of the class group of $R$ being torsion.