Editing Commutative ring (section)

=== Spectrum ===
{{Main|Spectrum of a ring}}
[[Image:Spec Z.png|right|400px|thumb|Spec ('''Z''') contains a point for the zero ideal. The closure of this point is the entire space. The remaining points are the ones corresponding to ideals (''p''), where ''p'' is a prime number. These points are closed.]]
The ''spectrum of a ring'' <math>R</math>,{{efn|This notion can be related to the [[Spectrum of an operator|spectrum]] of a linear operator; see ''[[Spectrum of a C*-algebra]]'' and ''[[Gelfand representation]]''.}} denoted by <math>\text{Spec}\ R</math>, is the set of all prime ideals of <math>R</math>. It is equipped with a topology, the [[Zariski topology]], which reflects the algebraic properties of <math>R</math>: a basis of open subsets is given by
<math display="block">D\left(f\right) = \left\{p \in \text{Spec} \ R,f \not\in p\right\},</math> where <math>f</math> is any ring element.
Interpreting <math>f</math> as a function that takes the value ''f'' mod ''p'' (i.e., the image of ''f'' in the residue field ''R''/''p''), this subset is the locus where ''f'' is non-zero. The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps {{nowrap|''R'' → ''R''<sub>''f''</sub>}} and {{nowrap|''R'' → ''R'' / ''fR''}} correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary [[open immersion|open]] and [[closed immersion]]s respectively. Even for basic rings, such as illustrated for {{nowrap|1=''R'' = '''Z'''}} at the right, the Zariski topology is quite different from the one on the set of real numbers.

The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec (''R''). For an [[algebraically closed field]] ''k'', mSpec (k[''T''<sub>1</sub>, ..., ''T''<sub>''n''</sub>] / (''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>)) is in bijection with the set
{{block indent|1= {''x'' =(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) ∊ ''k''<sup>''n''</sup> | ''f''<sub>1</sub>(''x'') = ... = ''f''<sub>''m''</sub>(''x'') = 0.} }}
Thus, maximal ideals reflect the geometric properties of solution sets of polynomials, which is an initial motivation for the study of commutative rings. However, the consideration of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimal prime ideals (i.e., the ones not strictly containing smaller ones) correspond to the [[irreducible component]]s of Spec ''R''. For a Noetherian ring ''R'', Spec ''R'' has only finitely many irreducible components. This is a geometric restatement of [[primary decomposition]], according to which any ideal can be decomposed as a product of finitely many [[primary ideal]]s. This fact is the ultimate generalization of the decomposition into prime ideals in Dedekind rings.