A prime ideal of S−1R[X] is the extension of a unique prime ideal of R, so that the morphism Spec(S−1R[X])toSpec(R) is a bijection, and even an homeomorphism. All the extensions of residual fields induced are pure transcendental of transcendence degre 1.
As an example, if you look at the case R=mathbbZ, the morphism of schemes you get "puts in family" the extensions of fields mathbbFphookrightarrowmathbbFp(X).
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