View source for Spectrum of a ring

<!--
s0865901.png
$#A+1 = 71 n = 0
$#C+1 = 71 : ~/encyclopedia/old_files/data/S086/S.0806590 Spectrum of a ring
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A topological space  $   \mathop{\rm Spec}  A $
whose points are the prime ideals  $  \mathfrak p $
of a ring  $  A $
with the [[Zariski topology|Zariski topology]] (also called the spectral topology). It is assumed that  $  A $
is commutative and has an identity. The elements of  $  A $
can be regarded as functions on  $   \mathop{\rm Spec}  A $
by setting  $  a( \mathfrak p ) \equiv a $
$   \mathop{\rm mod}  \mathfrak p \in A / \mathfrak p $. 
$   \mathop{\rm Spec}  A $
supports a sheaf of local rings  $  {\mathcal O} (  \mathop{\rm Spec}  A) $, 
called its structure sheaf. For a point  $  \mathfrak p \in  \mathop{\rm Spec}  A $, 
the stalk of  $  {\mathcal O} (  \mathop{\rm Spec}  A ) $
over  $  \mathfrak p $
is the localization  $  A _ {\mathfrak p} $
of  $  A $
at  $  \mathfrak p $.

To any identity-preserving ring homomorphism  $  \phi :  A \rightarrow A  ^  \prime  $
there corresponds a continuous mapping  $  \phi  ^ {*} :   \mathop{\rm Spec}  A  ^  \prime  \rightarrow  \mathop{\rm Spec}  A $. 
If  $  N $
is the nil radical of  $  A $, 
then the natural mapping  $   \mathop{\rm Spec}  A /N \rightarrow  \mathop{\rm Spec}  A $
is a homeomorphism of topological spaces.

For a non-nilpotent element  $  f \in A $, 
let  $  D( f  )= (  \mathop{\rm Spec}  A ) \setminus  V( f  ) $, 
where  $  V( f  )= \{ {\mathfrak p \in  \mathop{\rm Spec}  A } : {f \in \mathfrak p } \} $. 
Then the ringed spaces  $  D( f  ) $
and  $   \mathop{\rm Spec}  A _ {(} f) $, 
where  $  A _ {(} f) $
is the localization of  $  A $
with respect to  $  f $, 
are isomorphic. The sets  $  D( f  ) $
are called the principal open sets. They form a basis for the topology on  $   \mathop{\rm Spec}  A $. 
A point  $  \mathfrak p \in  \mathop{\rm Spec}  A $
is closed if and only if  $  \mathfrak p $
is a maximal ideal of  $  A $. 
By assigning to  $  \mathfrak p $
its closure  $  \overline{ {\mathfrak p }}\; $
in  $   \mathop{\rm Spec}  A $, 
one obtains a one-to-one correspondence between the points of  $   \mathop{\rm Spec}  A $
and the set of closed irreducible subsets of  $   \mathop{\rm Spec}  A $. 
$   \mathop{\rm Spec}  A $
is quasi-compact, but usually not Hausdorff. The dimension of  $   \mathop{\rm Spec}  A $
is defined as the largest  $  n $
for which there is a sequence of distinct closed irreducible sets  $  Z _ {0} \subset  \dots \subset  Z _ {n} \subset   \mathop{\rm Spec}  A $.

Many properties of  $  A $
can be described in terms of  $   \mathop{\rm Spec}  A $. 
For example,  $  A/N $
is Noetherian if and only if  $   \mathop{\rm Spec}  A $
has the descending chain condition for closed sets;  $   \mathop{\rm Spec}  A $
is an irreducible space if and only if  $  A/N $
is an integral domain; the dimension of  $   \mathop{\rm Spec}  A $
coincides with the Krull dimension of  $  A $, 
etc.

Sometimes one considers the maximal spectrum  $   \mathop{\rm Specm}  A $, 
which is the subspace of  $   \mathop{\rm Spec}  A $
consisting of the closed points. For a graded ring  $  A $
one also considers the projective spectrum  $   \mathop{\rm Proj}  A $. 
If  $  A= \sum _ {n=} 0  ^  \infty  A _ {n} $, 
then the points of  $   \mathop{\rm Proj}  A $
are the homogeneous prime ideals  $  \mathfrak p $
of  $  A $
such that  $  \mathfrak p \Nps \sum _ {n=} 1  ^  \infty  A _ {n} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre commutative" , ''Eléments de mathématiques'' , '''XXVIII''' , Hermann (1961) {{MR|0217051}} {{MR|0171800}} {{ZBL|0119.03603}} {{ZBL|0108.04002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>

====Comments====
The continuous mapping  $  \phi  ^ {*} :   \mathop{\rm Spec}  A  ^  \prime  \rightarrow  \mathop{\rm Spec}  A $
defined by a unitary ring homomorphism  $  \phi :  A \rightarrow A  ^  \prime  $
is given by  $  \phi  ^ {*} ( \mathfrak p  ^  \prime  ) = \phi  ^ {-} 1 ( \mathfrak p  ^  \prime  ) $.

The pair  $  (  \mathop{\rm Spec}  A, {\mathcal O} (  \mathop{\rm Spec}  A )) $
is an [[Affine scheme|affine scheme]].

Similarly,  $   \mathop{\rm Proj}  A $
supports a sheaf of local rings  $  {\mathcal O} (  \mathop{\rm Proj}  A) $, 
the stalk of which at a point  $  \mathfrak p $
is the homogeneous localization  $  A _ {( \mathfrak p ) }  $
of  $  A $
at  $  \mathfrak p $. 
(See also [[Localization in a commutative algebra|Localization in a commutative algebra]].) The pair  $  (  \mathop{\rm Proj}  A, {\mathcal O} (  \mathop{\rm Proj}  A )) $
is a [[Projective scheme|projective scheme]].

Spectra have also been studied for non-commutative rings, cf. [[#References|[a1]]].

For Krull dimension see [[Dimension|Dimension]] (of an associative ring).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. van Oystaeyen, A. Verschoren, "Non-commutative algebraic geometry" , ''Lect. notes in math.'' , '''887''' , Springer (1981) {{MR|639153}} {{ZBL|0477.16001}} </TD></TR></table>