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

Abstract EN

Let R ⊂ S be an extension of commutative rings, with X an indeterminate, such that the extension R X ⊂ S X of Nagata rings has FIP (i.e., S X has only finitely many R X -subalgebras).

Then, the number of R X -subalgebras of S X equals the number of R-subalgebras of S.

In fact, the function from the set of R-subalgebras of S to the set of R X -subalgebras of S X given by T ↦ T X is an order-isomorphism.