Editing Spectrum of a ring (section)

== Motivation from algebraic geometry ==

Following on from the example, in [[algebraic geometry]] one studies ''algebraic sets'', i.e. subsets of <math>K^n</math> (where <math>K</math> is an [[algebraically closed field]]) that are defined as the common zeros of a set of [[polynomial]]s in <math>n</math> variables. If <math>A</math> is such an algebraic set, one considers the commutative ring <math>R</math> of all [[polynomial function]]s <math>A\to K</math>. The ''maximal ideals'' of <math>R</math> correspond to the points of <math>A</math> (because <math>K</math> is algebraically closed), and the ''prime ideals'' of <math>R</math> correspond to the ''irreducible subvarieties'' of <math>A</math> (an algebraic set is called [[irreducible set|irreducible]] if it cannot be written as the union of two proper algebraic subsets).

The spectrum of <math>R</math> therefore consists of the points of <math>A</math> together with elements for all irreducible subvarieties of <math>A</math>. The points of <math>A</math> are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of <math>A</math>, i.e. the maximal ideals in <math>R</math>, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in <math>R</math>, i.e. <math>\operatorname{MaxSpec}(R)</math>, together with the Zariski topology, is [[homeomorphic]] to <math>A</math> also with the Zariski topology.

One can thus view the topological space <math>\operatorname{Spec}(R)</math> as an "enrichment" of the topological space <math>A</math> (with Zariski topology): for every irreducible subvariety of <math>A</math>, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding irreducible subvariety. One thinks of this point as the [[generic point]] for the irreducible subvariety. Furthermore, the structure sheaf on <math>\operatorname{Spec}(R)</math> and the sheaf of polynomial functions on <math>A</math> are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with the Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of [[scheme (mathematics)|schemes]].