Ring Spectrum

The spectrum of a ring is the set of proper prime ideals,

(1)

The classical example is the spectrum of polynomial rings. For instance,

(2)

and

(3)

The points are, in classical algebraic geometry, algebraic varieties. Note that are maximal ideals, hence also prime.

The spectrum of a ring has a topology called the Zariski topology. The closed sets are of the form

(4)

For example,

(5)

Every prime ideal is closed except for , whose closure is .