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.
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