COMMUTATIVE RINGS WHOSE PRIME IDEALS ARE RADICALLY PERFECT


Erdogdu V., Harman S.

JOURNAL OF COMMUTATIVE ALGEBRA, vol.5, no.4, pp.527-544, 2013 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 5 Issue: 4
  • Publication Date: 2013
  • Doi Number: 10.1216/jca-2013-5-4-527
  • Title of Journal : JOURNAL OF COMMUTATIVE ALGEBRA
  • Page Numbers: pp.527-544

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 [10, 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.