Commutative Algebra/Spectrum with Zariski topology

On $$\operatorname{Spec} R$$, we will define a topology, turning $$\operatorname{Spec} R$$ 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.