When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring

David E Dobbs,Martine Picavet-L’Hermitte,Gabriel Picavet

doi:10.1155/2014/315919

Abstract

LetR⊂Sbe an extension of commutative rings, withXan indeterminate, such that the extensionRX⊂SXof Nagata rings has FIP (i.e.,SXhas only finitely manyRX-subalgebras). Then, the number ofRX-subalgebras ofSXequals the number ofR-subalgebras ofS. In fact, the function from the set ofR-subalgebras ofSto the set ofRX-subalgebras ofSXgiven byT ↦TXis an order-isomorphism.

Highlights

Introduction and NotationAll rings considered below are commutative and unital; all inclusions of rings and all ring homomorphisms are unital
If I is an ideal of a ring R, V(I) := VR(I) := {P ∈ Spec(R) | I ⊆ P}; and |Y| denotes the cardinality of a set Y
By a chain of R-subalgebras of S, we mean a set of elements of [R, S] that are pairwise comparable with respect to inclusion

Summary

Introduction and Notation

All rings considered below are commutative and unital; all inclusions of rings and all ring homomorphisms are unital. Evidence for Theorem 32 was provided in [4, Propositions 4.4, 4.14, 4.17], where it was shown that if R ⊆ S has FIP, φ is an order-isomorphism in the following three cases: R ⊆ S is an integrally closed extension; R ⊆ S is a subintegral extension such that R(X) ⊂ S(X) has FIP; R ⊆ S is a seminormal infraintegral extension. This fact is used in the proof of Theorem 12, which obtains an affirmative answer to our main question in case of an arbitrary integral t-closed extension R ⊆ S. By a characterization of integral FCP extensions in [1, Theorem 4.2(a)], (R : S) is an intersection of finitely many, say, m, pairwise distinct maximal ideals of S. K ⊂ R is t-closed if and only if R is a field

T-Closed FIP Extensions of Nagata Rings

The General Case