Does quotient commute with localization?

Let $R$ be a commutative ring, and $I \subset R$ an ideal. If we choose an element $x \in R$ we can consider $(R/I)_x$ and $R_x/I_x$. In general, does localization commute with quotient? i.e. $(R/I)_x \simeq R_x/I_x$? If not... are there hypotheses on $x$, $R$ or $I$ under which localization commutes with quotient?

Solution 1:

In general, if $$0\to M\to N\to P\to0$$ is a short exact sequence of $R$-modules and $S$ is a multiplicative set in $R$, the localized sequence $$0\to M_S\to N_S\to P_S\to0$$ is also exact.

If $I\subseteq R$ is an ideal, we have a short exact sequence $$0\to I\to R\to R/I\to0$$ of $R$-modules, and therefore for $x\in R$ we get that $$0\to I_x\to R_x\to (R/I)_x\to0$$ is also exact. This means, among other things, that $(R/I)_x$ is isomorphic to $R_x/I_x$.

This isomorphism is as a $R$-module, and you probably want it to be as rings: that's a little extra work: the above map, which we have just observed to be a bijection, is actually a map of rings —checking this is simply a matter of writing down what it means.