TITLE: Localizing at the primitive polynomials?
QUESTION [11 upvotes]: 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?

REPLY [3 votes]: Google "Kronecker function ring". This is the germ of the idea behind Kronecker's divisor theory. An elementary historically-minded introduction can be found in Harold Edward's book "Divisor Theory".