For any UFD $R$, the concept of a primitive polynomial (gcd of the coefficients is 1) makes sense in $R[x]$. The product of two primitive polynomials is primitive (Gauss's Lemma), and certainly 1 is a primitive polynomial, so the primitive polynomials form a multiplicative subset $S$ of $R[x]$  hence we can form the ring $S^{1}R[x]$. What can we say about it? What does this look like geometrically?
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 $\mathbb{F}_p\hookrightarrow\mathbb{F}_p(X)$. 


Google "Kronecker function ring". This is the germ of the idea behind Kronecker's divisor theory. An elementary historicallyminded introduction can be found in Harold Edward's book "Divisor Theory". 

