Galois correspondence between a ring and its max-spectrum

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

Definition

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

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

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

The Galois correspondence

This binary relation induces a Galois correspondence as follows:

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

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

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

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

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

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

The closed sets on both sides

• A set of the $R$-side is closed iff it occurs as an intersection of maximal ideals.
• The closed sets on the $Max-Spec(R)$-side correspond to the collections of all maximal ideals containing a given ideal (without loss of generality, we may assume that that ideal is an intersection of maximal ideals). In other words, for every intersection of maximal ideals, there is a corresponding closed set in $Max-Spec(R)$, and this closed set is the set of all prime ideals containing that ideal.

Note that, in general, every radical ideal may not be expressible as an intersection of maximal ideals (although every intersection of maximal ideals is a radical ideal). Thus, the closed sets of $Spec(R)$ need not be completely determined by their intersections with $Max-Spec(R)$. A ring for which every radical ideal is an intersection of maximal ideals is termed a Jacobson ring.

It turns out that this definition of closed sets turns $Max-Spec(R)$ 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. Moroever the topology that we get on $Max-Spec(R)$ is the same as the subspace topology that we get from the natural topology on the spectrum of $R$.