Commutative Algebra/Spectrum with Zariski topology

On $\mathrm{Spec}R$, we will define a topology, turning $\mathrm{Spec}R$ into a topological space. This topology will be called Zariski topology, although only Alexander Grothendieck gave the definition in the above generality.

Closed sets

The sets $V(S)$, where $S$ ranges over subsets of $R$, satisfy the following equations:

Proof:

The first two items are straightforward. For the third, we use induction on $n$. $n=1$ is clear; otherwise, the direction $\subseteq$ is clear, and the other direction follows from lemma 14.20.$◻$