The Stacks project

Example 10.35.15. Here is a simple example that shows Lemma 10.35.14 to be false if $R$ is not Jacobson. Consider the ring $R = \mathbf{Z}_{(2)}$, i.e., the localization of $\mathbf{Z}$ at the prime $(2)$. The localization of $R$ at the element $2$ is isomorphic to $\mathbf{Q}$, in a formula: $R_2 \cong \mathbf{Q}$. Clearly the map $R \to R_2$ maps the closed point of $\mathop{\mathrm{Spec}}(\mathbf{Q})$ to the generic point of $\mathop{\mathrm{Spec}}(R)$.