Editing Commutative algebra (section)

==Main tools and results==

===Noetherian rings===
{{Main|Noetherian ring}}
A '''Noetherian ring''', named after [[Emmy Noether]], is a ring in which every  [[ideal (ring theory)|ideal]] is [[finitely generated ideal|finitely generated]]; that is, all elements of any ideal can be written as a [[linear combination]]s of a finite set of elements, with coefficients in the ring. 

Many commonly considered commutative rings are Noetherian, in particular, every [[field (mathematics)|field]], the ring of the [[integer]], and every [[polynomial ring]] in one or several indeterminates over them. The fact that polynomial rings over a field are Noetherian is called [[Hilbert's basis theorem]].

Moreover, many ring constructions preserve the Noetherian property. In particular, if a commutative ring {{math|R}} is Noetherian, the same is true for every polynomial ring over it, and for every [[quotient ring]], [[localization (commutative algebra)|localization]], or [[completion of a ring|completion]] of the ring.

The importance of the Noetherian property lies in its ubiquity and also in the fact that many important theorems of commutative algebra require that the involved rings are Noetherian, This is the case, in particular of [[Lasker–Noether theorem]], the [[Krull intersection theorem]], and [[Nakayama's lemma]].

Furthermore, if a ring is Noetherian, then it satisfies the [[descending chain condition]] on [[prime ideal]]s, which implies that every  Noetherian [[local ring]] has a finite [[Krull dimension]].

===Primary decomposition===
{{Main|Primary decomposition}}
An ideal ''Q'' of a ring is said to be ''[[Primary ideal|primary]]'' if ''Q'' is [[proper subset|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y<sup>n</sup>'' ∈ ''Q'' for some positive integer ''n''. In '''Z''', the primary ideals are precisely the ideals of the form (''p<sup>e</sup>'') where ''p'' is prime and ''e'' is a positive integer. Thus, a primary decomposition of (''n'') corresponds to representing (''n'') as the intersection of finitely many primary ideals.

The ''[[Lasker–Noether theorem]]'', given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:

{{math theorem|name=Lasker-Noether Theorem
|math_statement=Let ''R'' be a commutative Noetherian ring and let ''I'' be an ideal of ''R''. Then ''I'' may be written as the intersection of finitely many primary ideals with distinct [[Radical of an ideal|radicals]]; that is:

: <math>I=\bigcap_{i=1}^t Q_i</math>

with ''Q<sub>i</sub>'' primary for all ''i'' and Rad(''Q<sub>i</sub>'') ≠ Rad(''Q<sub>j</sub>'') for ''i'' ≠ ''j''. Furthermore, if:

: <math>I=\bigcap_{i=1}^k P_i</math>

is decomposition of ''I'' with Rad(''P<sub>i</sub>'') ≠ Rad(''P<sub>j</sub>'') for ''i'' ≠ ''j'', and both decompositions of ''I'' are ''irredundant'' (meaning that no proper subset of either {''Q''<sub>1</sub>, ..., ''Q<sub>t</sub>''} or {''P''<sub>1</sub>, ..., ''P<sub>k</sub>''} yields an intersection equal to ''I''), ''t'' = ''k'' and (after possibly renumbering the ''Q<sub>i</sub>'') Rad(''Q<sub>i</sub>'') = Rad(''P<sub>i</sub>'') for all ''i''.}}

For any primary decomposition of ''I'', the set of all radicals, that is, the set {Rad(''Q''<sub>1</sub>), ..., Rad(''Q<sub>t</sub>'')} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the [[associated prime|assassinator]] of the module ''R''/''I''; that is, the set of all [[annihilator (ring theory)|annihilators]] of ''R''/''I'' (viewed as a module over ''R'') that are prime.

===Localization===
{{Main|Localization (algebra)}}

The [[localization (algebra)|localization]] is a formal way to introduce the "denominators" to a given ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of [[algebraic fraction|fractions]] 
:<math>\frac{m}{s}</math>.
where the [[denominator]]s ''s'' range in a given subset ''S'' of ''R''. The archetypal example is the construction of the ring '''Q''' of rational numbers from the ring '''Z''' of integers.

===Completion===
{{Main|Completion (ring theory)}}
A [[completion (ring theory)|completion]] is any of several related [[functor]]s on [[ring (mathematics)|ring]]s and [[module (mathematics)|modules]] that result in complete [[topological ring]]s and modules. Completion is similar  to [[localization of a ring|localization]], and together they are among the most basic tools in analysing [[commutative ring]]s. Complete commutative rings have simpler structure than the general ones and [[Hensel's lemma]] applies to them.

===Zariski topology on prime ideals===
{{Main|Zariski topology}}
The [[Zariski topology]] defines a [[topological space|topology]] on the [[spectrum of a ring]] (the set of prime ideals).<ref>{{cite book
| last1 = Dummit
| first1 = D. S.
| last2 = Foote
| first2 = R.
| title = Abstract Algebra
| url = https://archive.org/details/abstractalgebra00dumm_304
| url-access = limited
| publisher = Wiley
| pages = [https://archive.org/details/abstractalgebra00dumm_304/page/n84 71]–72
| year = 2004
| edition = 3
| isbn = 9780471433347 
}}</ref>  In this formulation, the Zariski-closed sets are taken to be the sets

:<math>V(I) = \{P \in \operatorname{Spec}\,(A) \mid I \subseteq P\}</math>

where ''A'' is a fixed commutative ring and ''I'' is an ideal.  This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations . To see the connection with the classical picture, note that for any set ''S'' of polynomials (over an algebraically closed field), it follows from [[Hilbert's Nullstellensatz]] that the points of ''V''(''S'') (in the old sense) are exactly the tuples (''a<sub>1</sub>'', ..., ''a<sub>n</sub>'') such that the ideal (''x<sub>1</sub>'' - ''a<sub>1</sub>'', ..., ''x<sub>n</sub>'' - ''a<sub>n</sub>'') contains ''S''; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.  Thus, ''V''(''S'') is "the same as" the maximal ideals containing ''S''.  Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.