Possible Characterization of localization of $\mathbb{Q}[x, y]$

Question

So I wish to know if it is possible to Characterize the localization of $\mathbb{Q}[x, y]$. The motivation behind this question is that I was asked to determine (True or false) that every subring of $\mathbb{Q}(x,y)$ containing $\mathbb{Q}[x, y]$ is some form of localization. I believe it to be false; and was hoping that some knowing some form of characterization can help me in coming up with a counter-example.

While I'm at it; any insights behind my question is appreciated.

Cheers

Answer 1

Every nontrivial localization of a ring introduces new units. Your goal is to find a ring extension $\Bbb Q[x,y]\subset R \subset\Bbb Q(x,y)$ so that $R$ has no units other than $\Bbb Q$. Hint: blowups. A full solution is under the spoiler.

Consider $\Bbb Q[x,y,x/y]$ - this is the coordinate algebra of one of the affine patches of the blowup of $\Bbb A_\Bbb Q^2$ at the origin. It fits the role of $R$ from before, but it has no new units: it's isomorphic to the polynomial ring on the variables $y,x/y$, and the only units in a polynomial ring are the invertible constants.