The spectrum of a ring (statement)

Definition. $\operatorname{Spec}(R)=\{\mathfrak{p}\subseteq R:\mathfrak{p}\text{ prime ideal}\}$.

Zariski topology. Closed sets are $V(I)=\{\mathfrak{p}:\mathfrak{p}\supseteq I\}$ for ideals $I\subseteq R$.

Example. $\operatorname{Spec}(\mathbb{Z})=\{(0)\}\cup\{(p):p\text{ prime}\}$.

Topological structure. The point $(0)$ is the generic point, dense in $\operatorname{Spec}(\mathbb{Z})$. The points $(p)$ are closed.

Conclusion. $\operatorname{Spec}(\mathbb{Z})$ consists of the zero ideal and the prime ideals $(p)$, with Zariski topology where $(0)$ is dense and each $(p)$ is a closed point.