prime spectrum

1 Spec as a set

Let R be any commutative ring with identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The prime spectrum Spec(R) of R is defined to be the set

{PRP is a prime ideal of R}.

For any subset A of R, we define the varietyMathworldPlanetmathPlanetmathPlanetmath of A to be the set


It is enough to restrict attention to subsets of R which are ideals, since, for any subset A of R, we have V(A)=V(I) where I is the ideal generated byPlanetmathPlanetmath A. In fact, even more is true: V(I)=V(I) where I denotes the radicalPlanetmathPlanetmathPlanetmath of the ideal I.

2 Spec as a topological space

We impose a topologyMathworldPlanetmath on Spec(R) by defining the sets V(A) to be the collectionMathworldPlanetmath of closed subsets of Spec(R) (that is, a subset of Spec(R) is open if and only if it equals the complementPlanetmathPlanetmath of V(A) for some subset A). The equations

αV(Iα) = V(αIα)
i=1nV(Ii) = V(i=1nIi),

for any ideals Iα, Ii of R, establish that this collection does constitute a topology on Spec(R). This topology is called the Zariski topologyMathworldPlanetmath in light of its relationship to the Zariski topology on an algebraic variety (see SectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 4 below). Note that a point PSpec(R) is closed if and only if PR is a maximal idealMathworldPlanetmath.

A distinguished open set of Spec(R) is defined to be an open set of the form


for any element fR. The collection of distinguished open sets forms a topological basis for the open sets of Spec(R). In fact, we have


The topological space Spec(R) has the following additional properties:

  • Spec(R) is compactPlanetmathPlanetmath (but almost never HausdorffPlanetmathPlanetmath).

  • A subset of Spec(R) is an irreduciblePlanetmathPlanetmathPlanetmathPlanetmath closed set if and only if it equals V(P) for some prime idealMathworldPlanetmathPlanetmath P of R.

  • For fR, let Rf denote the localizationMathworldPlanetmath of R at f. Then the topological spaces Spec(R)f and Spec(Rf) are naturally homeomorphicMathworldPlanetmath, via the correspondence sending a prime ideal of R not containing f to the induced prime ideal in Rf.

  • For PSpec(R), let RP denote the localization of R at the prime ideal P. Then the topological spaces V(P)Spec(R) and Spec(RP) are naturally homeomorphic, via the correspondence sending a prime ideal of R contained in P to the induced prime ideal in RP.

3 Spec as a sheaf

For convenience, we adopt the usual convention of writing X for Spec(R). For any fR and PXf, let ιf,P:RfRP be the natural inclusion mapMathworldPlanetmath. Define a presheafMathworldPlanetmathPlanetmathPlanetmath of rings 𝒪X on X by setting

U has an open cover { X f α } with elements s α R f α
such that = s P ι f α , P ( s α ) whenever P X f α

for each open set UX. The restrictionPlanetmathPlanetmath map resU,V:𝒪X(U)𝒪X(V) is the map induced by the projection map


for each open subset VU. The presheaf 𝒪X satisfies the following properties:

  1. 1.

    𝒪X is a sheaf.

  2. 2.

    𝒪X(Xf)=Rf for every fR.

  3. 3.

    The stalk (𝒪X)P is equal to RP for every PX. (In particular, X is a locally ringed space.)

  4. 4.

    The restriction sheaf of 𝒪X to Xf is isomorphicPlanetmathPlanetmathPlanetmath as a sheaf to 𝒪Spec(Rf).

4 Relationship to algebraic varieties

Spec(R) is sometimes called an affine schemeMathworldPlanetmath because of the close relationship between affine varietiesMathworldPlanetmath in 𝔸kn and the Spec of their corresponding coordinate rings. In fact, the correspondence between the two is an equivalence of categories, although a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath statement of this equivalence requires the notion of morphisms of schemes and will not be given here. Nevertheless, we explain what we can of this correspondence below.

Let k be a field and write as usual 𝔸kn for the vector space kn. Recall that an affine variety V in 𝔸kn is the set of common zeros of some prime ideal Ik[X1,,Xn]. The coordinate ring of V is defined to be the ring R:=k[X1,,Xn]/I, and there is an embeddingMathworldPlanetmathPlanetmath i:VSpec(R) given by


The function i is not a homeomorphism, because it is not a bijection (its image is contained inside the set of maximal ideals of R). However, the map i does define an order preserving bijection between the open sets of V and the open sets of Spec(R) in the Zariski topology. This isomorphismMathworldPlanetmathPlanetmathPlanetmath between these two lattices of open sets can be used to equate the sheaf Spec(R) with the structure sheaf of the variety V, showing that the two objects are identical in every respect except for the minor detail of Spec(R) having more points than V.

The additional points of Spec(R) are valuable in many situations and a systematic study of them leads to the general notion of schemes. As just one example, the classical Bezout’s theorem is only valid for algebraically closed fields, but admits a scheme–theoretic generalizationPlanetmathPlanetmath which holds over non–algebraically closed fields as well. We will not attempt to explain the theory of schemes in detail, instead referring the interested reader to the references below.

Remark. The spectrum Spec(R) of a ring R may be generalized to the case when R is not commutativePlanetmathPlanetmathPlanetmath, as long as R contains the multiplicative identityPlanetmathPlanetmath. For a ring R with 1, the Spec(R), like above, is the set of all proper prime ideals of R. This definition is used to develop the noncommutative version of Hilbert’s Nullstellensatz.


  • 1 Robin Hartshorne, Algebraic GeometryMathworldPlanetmathPlanetmath, Springer–Verlag New York, Inc., 1977 (GTM 52).
  • 2 David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, Springer–Verlag, 1999 (LNM 1358).
  • 3 Louis H. Rowen, Ring Theory, Vol. 1, Academic Press, New York, 1988.
Title prime spectrum
Canonical name PrimeSpectrum
Date of creation 2013-03-22 12:38:07
Last modified on 2013-03-22 12:38:07
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Definition
Classification msc 14A15
Related topic Scheme
Related topic Sheaf
Related topic ZariskiTopology
Related topic LocallyRingedSpace
Defines distinguished open set