Localization preserves Noetherian rings

• Since $\mathbb Z$ is Noetherian, its localization $\mathbb Z_{(p)}$ is Noetherian for every prime $p$.
• If $R=k[x,y]$ and $S=\{1,x,x^2,\ldots\}$, then $S^{-1}R=R_x$ is Noetherian.

The converse fails: a localization can be Noetherian even when its source is not. Let $R=k[x_1,x_2,\ldots]$, which has the infinite ascending chain $(x_1)\subsetneq(x_1,x_2)\subsetneq\cdots$ of ideals, and localize at all nonzero elements. The resulting fraction field $\operatorname{Frac}(R)$ is a field, hence Noetherian. The preservation theorem is often used together with exactness of localization.