Definition:Zariski Topology/Spectrum of Ring

Definition
Let $A$ be a commutative ring with unity.

Let $\operatorname{Spec} \left({A}\right)$ be the prime spectrum of $A$.

The Zariski topology is that with closed sets:
 * $V \subseteq \operatorname{Spec} \left({A}\right)$

such that for some $S \subseteq A$:
 * $V = \left\{{\mathfrak p \in \operatorname{Spec} \left({A}\right): \mathfrak p \supseteq S}\right\}$

We usually write $V = V \left({S}\right)$, and specify $S$ to determine the set $V$.

Also see

 * Zariski Topology is Topology
 * Definition:Zariski Topology on Maximal Spectrum of Ring