Galois correspondence between a ring and its spectrum

This article defines a Galois correspondence induced by a binary relation, the binary relation here being: containment of an element in a prime ideal

Definition

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. Further information: intersection of prime equals radical
• 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 ${\mathcal {Z}}(I)$.