Max-spectrum of a commutative unital ring

Definition

The max-spectrum of a commutative unital ring is a highly structured object that captures much of the geometry associated with the ring. We here describe its structure at various levels.

Set-theoretic structure

Set-theoretically, the spectrum is the set of maximal ideals in the ring. In this sense, it is a subset of the spectrum, which is the set of all prime ideals.

Topological structure

A subset in the max-spectrum is deemed a closed set if and only if there exists an intersection of maximal ideals, such that the given subset is the set of all maximal ideals, containing the given intersection of maximal ideals.

This topology is sometimes termed the Zariski topology though it should not be confused with the Zariski topology for a polynomial ring over an algebraically closed field.

The topology described above on the max-spectrum can be derived very naturally from the Galois correspondence between a ring and its max-spectrum: the closed sets in the topology are the same as the sets which are closed in the sense of the Galois correspondence. It is also the same as the subspace topology arising from the topology we put on the spectrum.