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

Let R⊂S be an extension of commutative rings, with X an indeterminate, such that the extension RX⊂SX of Nagata rings has FIP (i.e., SX has only finitely many RX-subalgebras). Then, the number of RX-subalgebras of SX equals the number of R-subalgebras of S. In fact, the function from the set of R-subalgebras of S to the set of RX-subalgebras of SX given by T ↦TX is an order-isomorphism