Editing Noetherian ring (section)

== Key theorems ==
Many important theorems in ring theory (especially the theory of [[commutative ring]]s) rely on the assumptions that the rings are Noetherian.

===Commutative case===
*Over a commutative Noetherian ring, each ideal has a [[primary decomposition]], meaning that it can be written as an [[intersection (set theory)|intersection]] of finitely many [[primary ideal]]s (whose [[radical of an ideal|radical]]s are all distinct) where an ideal ''Q'' is called primary if it is [[proper ideal|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y''<sup>&thinsp;''n''</sup> ∈ ''Q'' for some positive integer ''n''. For example, if an element <math>f = p_1^{n_1} \cdots p_r^{n_r}</math> is a product of powers of distinct prime elements, then <math>(f) = (p_1^{n_1}) \cap \cdots \cap (p_r^{n_r})</math> and thus the primary decomposition is a direct generalization of [[prime factorization]] of integers and polynomials.<ref>{{harvnb|Eisenbud|1995|loc=Proposition 3.11.}}</ref>
*A Noetherian ring is defined in terms of ascending chains of ideals. The [[Artin–Rees lemma]], on the other hand, gives some information about a descending chain of ideals given by powers of ideals <math>I \supseteq I^2 \supseteq I^3 \supseteq \cdots </math>. It is a technical tool that is used to [[mathematical proof|prove]] other key theorems such as the [[Krull intersection theorem]].
*The [[dimension theory (algebra)|dimension theory]] of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem, [[Krull's principal ideal theorem]], already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian) [[universally catenary ring]]s, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.

===Non-commutative case===
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*[[Goldie's theorem]]