Galois correspondence between a ring and its spectrum

The binary relation
Let $$R$$ be a commutative unital ring, and denote by $$Spec(R)$$ the set of prime ideals in $$R$$ (also termed the spectrum of $$R$$). Define the following binary relation $$B$$ between $$R$$ and $$Spec(R)$$:

$$(a,I) \in B \iff a \in I$$

In other words, an element of $$R$$ is related to an element of $$Spec(R)$$ iff the element of $$R$$ lies in that prime ideal.

The Galois correspondence
This binary relation induces a Galois correspondence as follows:


 * Let $$\mathcal{Z}$$ be a map from the collection of subsets of $$R$$, to the collection of subsets of $$Spec(R)$$, defined as:

$$\mathcal{Z}(X) = \{ I \in Spec(R) | x \in I \ \forall \ x \in X \}$$

In other words, $$\mathcal{Z}(X)$$ is the collection of all prime ideals containing $$X$$.


 * Let $$\mathcal{I}$$ be a map from the collection of subsets of $$Spec(R)$$ to the collection of subsets of $$R$$, defined as:

$$\mathcal{I}(Y) = \{ x \in R | x \in I \ \forall \ I \in Y \}$$

In other words, $$\mathcal{I}(Y)$$ is the set of elements which lie in the intersection of all the ideals in $$Y$$.

The closed sets on both sides

 * The closed sets on the $$R$$-side are the radical ideals. To see this, note that a closed set on the ring-side must be a subset which arises as an intersection of prime ideals. This is precisely the same as being a radical ideal.
 * The closed sets on the $$Spec(R)$$-side correspond to the collections of all prime ideals containing a given radical ideal. In other words, for every radical ideal, there is a corresponding closed set in $$Spec(R)$$, and this closed set is the set of all prime ideals containing that radical ideal.

It turns out that this definition of closed sets turns $$Spec(R)$$ into a topological space. Note that the whole space being closed, and an arbitrary intersection of closed sets being closed, follows directly from the properties of a Galois correspondence. However, the fact that the empty set is closed, and that a finite union of closed sets is closed, uses attributes particular to ring theory and to prime ideals. We usually study $$Spec(R)$$ with this topology.

In particular, if we replace $$Spec(R)$$ by the collection of all ideals and define an analogous Galois correspondence, we do not get a topology on the set of all prime ideals.

Contravariance
The Galois correspondence between a ring and its spectrum is a contravariant correspondence, in the following sense. Suppose $$f:R \to S$$ is a homomorphism of commutative unital rings. Then, we get a backward map from $$Spec(S)$$ to $$Spec(R)$$. The Galois correspondence commutes with this backward correspondence in the following sense:

If we start with a closed subset $$C$$ of $$Spec(S)$$, and take its image in $$Spec(R)$$, that is the same as the Galois correspondent to the inverse image of its Galois correspondent. In symbols:

$$\mathcal{Z}(f^{-1}\mathcal{I}(C)) = Spec(f)(C)$$

For quotients
When we take a quotient ring of $$R$$ by an ideal $$I$$, we get a subset of $$Spec(R)$$. More precisely $$Spec(R/I)$$ is naturally identified with the closed subset of $$Spec(R)$$ given as $$\mathcal{Z}(I)$$. Moreover, the Galois correspondence between $$R/I$$ and $$Spec(R/I)$$ is effectively the same as the Galois correspondence between subsets of $$R$$ which are unions of cosets of $$I$$, and subsets of $$Spec(R)$$ whichare contained in $$\mathcal{Z}(I)$$.