Efficient Generation of Prime Ideals in Polynomial Rings up to Radical


Erdogdu V.

COMMUNICATIONS IN ALGEBRA, vol.38, no.5, pp.1802-1807, 2010 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 5
  • Publication Date: 2010
  • Doi Number: 10.1080/00927870902917679
  • Title of Journal : COMMUNICATIONS IN ALGEBRA
  • Page Numbers: pp.1802-1807

Abstract

We call an ideal I of a ring R radically perfect if among all ideals 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. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the Jmax-radical of an ideal generated by two elements. We also show that if R is a Prufer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prufer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prufer domain with an extra condition imposed on it.