On the generalized principal ideal theorem and Krull domains

David Fenimore Anderson, David Earl Dobbs, Paul M. Eakin, Jr. and William James Heinzer

Vol. 146 (1990), No. 2, 201–215

DOI: 10.2140/pjm.1990.146.201

Abstract

If R ⊂ T is an integral extension of domains and R is Noetherian, then T satisfies (the conclusion of the) generalized principal ideal theorem (or GPIT for short). An example is given of a two-dimensional quasilocal domain R satisfying GPIT such that the integral closure of R is finite over R but does not satisfy GPIT. If a commutative ring R satisfies GPIT and an ideal I of R is generated by an R-sequence, then R∕I satisfies GPIT. If R is a Noetherian domain and G is a torsionfree abelian group, then R[G] satisfies GPIT. An example is given of a three-dimensional quasilocal Krull domain that does not satisfy GPIT because its maximal ideal is the radical of a 2-generated ideal.

Mathematical Subject Classification 2000

Primary: 13A15

Secondary: 13B22

Milestones

Received: 30 November 1988

Revised: 24 April 1989

Published: 1 December 1990