Spectrum of a ring
A topological space 
whose points are the prime ideals 
of a ring 
with the Zariski topology (also called the spectral topology). It is assumed that 
is commutative and has an identity. The elements of 
can be regarded as functions on 
by setting 
. 
supports a sheaf of local rings 
, called its structure sheaf. For a point 
, the stalk of 
over 
is the localization 
of 
at 
.
To any identity-preserving ring homomorphism 
there corresponds a continuous mapping 
. If 
is the nil radical of 
, then the natural mapping 
is a homeomorphism of topological spaces.
For a non-nilpotent element 
, let 
, where 
. Then the ringed spaces 
and 
, where 
is the localization of 
with respect to 
, are isomorphic. The sets 
are called the principal open sets. They form a basis for the topology on 
. A point 
is closed if and only if 
is a maximal ideal of 
. By assigning to 
its closure 
in 
, one obtains a one-to-one correspondence between the points of 
and the set of closed irreducible subsets of 
. 
is quasi-compact, but usually not Hausdorff. The dimension of 
is defined as the largest 
for which there is a sequence of distinct closed irreducible sets 
.
Many properties of 
can be described in terms of 
. For example, 
is Noetherian if and only if 
has the descending chain condition for closed sets; 
is an irreducible space if and only if 
is an integral domain; the dimension of 
coincides with the Krull dimension of 
, etc.
Sometimes one considers the maximal spectrum 
, which is the subspace of 
consisting of the closed points. For a graded ring 
one also considers the projective spectrum 
. If 
, then the points of 
are the homogeneous prime ideals 
of 
such that 
.
References
| [1] | N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , XXVIII , Hermann (1961) MR0217051 MR0171800 Zbl 0119.03603 Zbl 0108.04002 |
| [2] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Comments
The continuous mapping 
defined by a unitary ring homomorphism 
is given by 
.
The pair 
is an affine scheme.
Similarly, 
supports a sheaf of local rings 
, the stalk of which at a point 
is the homogeneous localization 
of 
at 
. (See also Localization in a commutative algebra.) The pair 
is a projective scheme.
Spectra have also been studied for non-commutative rings, cf. [a1].
For Krull dimension see Dimension (of an associative ring).
References
| [a1] | F. van Oystaeyen, A. Verschoren, "Non-commutative algebraic geometry" , Lect. notes in math. , 887 , Springer (1981) MR639153 Zbl 0477.16001 |
Spectrum of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_a_ring&oldid=15142