localization - Localizing at the primitive polynomials?

A prime ideal of $S^{-1}R[X]$ is the extension of a unique prime ideal of $R$, so that the morphism $Spec(S^{-1}R[X])to Spec(R)$ 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 $R=mathbb{Z}$, the morphism of schemes you get "puts in family" the extensions of fields $mathbb{F}_phookrightarrowmathbb{F}_p(X)$.