On the generalized principal ideal theorem and Krull domains. (English) Zbl 0746.13007

Anderson, David F.; Dobbs, David E.; Eakin, Paul M. jun.; Heinzer, William J.

Pac. J. Math. 146, No. 2, 201-215 (1990).

Summary: If \(R\subset T\) 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.

MSC:

13F10	Principal ideal rings
13G05	Integral domains

Keywords:

group ring; integral extension of domains; generalized principal ideal theorem; \(R\)-sequence; Noetherian domain