Kronecker Function Rings of Transcendental Field Extensions

Abstract

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all valuation rings of the extension F/D and investigate the ideal structure of Kr(F/D) in the case where D is an affine algebra over a subfield K of F and the extension F/K has countably infinite transcendence degree, by using the topological structure of the Zariski–Riemann space. We show that for any pair of nonnegative integers d and h, there are infinitely many prime ideals of dimension d and height h that are minimal over any proper nonzero finitely generated ideal of Kr(F/D).

Key Words:

• Kronecker function ring
• Prüfer domain
• Valuation

2000 Mathematics Subject Classification:

• Primary 13A15, 13F05, 13F30

ACKNOWLEDGMENTS

I am indebted to Bruce Olberding for helpful discussions and comments that helped to achieve the aim of this article. Also, I thank the referee for valuable comments and for pointing out errors in the previous versions of this article.

Notes

Communicated by R. Wiegand.