Editing Commutative ring (section)

== Spectrum of a commutative ring ==

=== Prime ideals ===
{{Main|Prime ideal}}
As was mentioned above, <math> \mathbb{Z} </math> is a [[unique factorization domain]]. This is not true for more general rings, as algebraists realized in the 19th century. For example, in
<math display="block">\mathbb{Z}\left[\sqrt{-5}\right]</math>
there are two genuinely distinct ways of writing 6 as a product:
<math display="block">6 = 2 \cdot 3 = \left(1 + \sqrt{-5}\right)\left(1 - \sqrt{-5}\right).</math>
Prime ideals, as opposed to prime elements, provide a way to circumvent this problem. A prime ideal is a proper (i.e., strictly contained in <math> R </math>) ideal <math> p </math> such that, whenever the product <math> ab </math> of any two ring elements <math> a </math> and <math> b </math> is in <math> p, </math> at least one of the two elements is already in <math> p .</math> (The opposite conclusion holds for any ideal, by definition.) Thus, if a prime ideal is principal, it is equivalently generated by a prime element. However, in rings such as <math>\mathbb{Z}\left[\sqrt{-5}\right],</math> prime ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any [[Dedekind ring]] (which includes <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> and more generally the [[algebraic integers|ring of integers in a number field]]) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals.

Any maximal ideal is a prime ideal or, more briefly, is prime. Moreover, an ideal <math>I</math> is prime if and only if the factor ring <math>R/I</math> is an integral domain. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. Yet another way of expressing the same is to say that the [[Complement (set theory)|complement]] <math>R \setminus p</math> is multiplicatively closed. The localisation <math>\left(R \setminus p\right)^{-1}R</math> is important enough to have its own notation: <math>R_p</math>. This ring has only one maximal ideal, namely <math>pR_p</math>. Such rings are called [[local ring|local]].

=== 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.

=== Affine schemes ===
The notion of a spectrum is the common basis of commutative algebra and [[algebraic geometry]]. Algebraic geometry proceeds by endowing Spec ''R'' with a [[sheaf (mathematics)|sheaf]] <math>\mathcal O</math> (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an [[affine scheme]]. Given an affine scheme, the underlying ring ''R'' can be recovered as the [[global section]]s of <math>\mathcal O</math>. Moreover, this one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any ''f'' : ''R'' → ''S'' gives rise to a [[continuous map]] in the opposite direction
{{block indent|1= Spec ''S'' → Spec ''R'', ''q'' ↦ ''f''<sup>−1</sup>(''q''), i.e. any prime ideal of ''S'' is mapped to its [[preimage]] under ''f'', which is a prime ideal of ''R''. }}
The resulting [[equivalence of categories|equivalence]] of the two said categories aptly reflects algebraic properties of rings in a geometrical manner.

Similar to the fact that [[manifold (mathematics)|manifolds]] are locally given by open subsets of '''R'''<sup>''n''</sup>, affine schemes are local models for [[scheme (mathematics)|schemes]], which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition.

=== Dimension ===
{{Main|Krull dimension}}
The ''Krull dimension'' (or dimension) dim ''R'' of a ring ''R'' measures the "size" of a ring by, roughly speaking, counting independent elements in ''R''. The dimension of algebras over a field ''k'' can be axiomatized by four properties:
* The dimension is a local property: {{nowrap|1=dim ''R'' = sup<sub>p ∊ Spec ''R''</sub> dim ''R''<sub>''p''</sub>}}.
* The dimension is independent of nilpotent elements: if {{nowrap|''I'' ⊆ ''R''}} is nilpotent then {{nowrap|1=dim ''R'' = dim ''R'' / ''I''}}.
* The dimension remains constant under a finite extension: if ''S'' is an ''R''-algebra which is finitely generated as an ''R''-module, then dim ''S'' = dim ''R''.
* The dimension is calibrated by dim {{nowrap|1=''k''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>] = ''n''}}. This axiom is motivated by regarding the polynomial ring in ''n'' variables as an algebraic analogue of [[affine space|''n''-dimensional space]].

The dimension is defined, for any ring ''R'', as the supremum of lengths ''n'' of chains of prime ideals
{{block indent|1= ''p''<sub>0</sub> ⊊ ''p''<sub>1</sub> ⊊ ... ⊊ ''p''<sub>''n''</sub>. }}
For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. The integers are one-dimensional, since chains are of the form (0) ⊊ (''p''), where ''p'' is a [[prime number]]. For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in [[commutative algebra]], the [[going-up theorem]] and [[Krull's principal ideal theorem]].