Localization of the polynomial ring $k[x]$.

Question

Forgive me if it is completely trivial, but what exactly does $k[x]_{(x)}$ mean? Is it the ring $k[x]$ localized by the ideal $I = (x)$ or localized at the polynomial $f(x) = x$?

Answer 1

It almost certainly means localization at the prime ideal $(x)$. That is, if we let $S=k[x]\smallsetminus (x)$, then $k[x]_{(x)}=S^{-1}k[x]$.

If somebody wants to write localization at the element $x$, i.e. taking $S=\{1,x,x^2,\dots\}$, then they will almost always write $k[x]_x$.

Answer 2

When you localize, you need a multiplicative subset. There are two options that this could mean, it could mean using $$ \{1,x,x^2,\cdots\} $$ or it could mean using $$ k[x]\setminus\langle x\rangle. $$

Usually, I would expect the first interpretation (in the types of things that I look at), but since $\langle x\rangle$ is a prime ideal, the other interpretation is possible.