Editing Noetherian ring (section)

== Properties ==
* If ''R'' is a Noetherian ring, then the [[polynomial ring]] <math>R[X]</math> is Noetherian by the [[Hilbert's basis theorem]]. By [[mathematical induction|induction]], <math>R[X_1, \ldots, X_n]</math> is a Noetherian ring. Also, {{math|''R''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>}}, the [[Formal power series|power series ring]], is a Noetherian ring.
* If {{math|''R''}} is a Noetherian ring and {{math|''I''}} is a two-sided ideal, then the [[quotient ring]] {{math|''R''/''I''}} is also Noetherian. Stated differently, the [[image (mathematics)|image]] of any [[surjective]] [[ring homomorphism]] of a Noetherian ring is Noetherian.
* Every finitely-generated [[commutative algebra (structure)|commutative algebra]] over a commutative Noetherian ring is Noetherian.  (This follows from the two previous properties.)
* A ring ''R'' is left-Noetherian [[if and only if]] every finitely generated left [[module (mathematics)|''R''-module]] is a [[Noetherian module]].
* If a commutative ring admits a [[faithful module|faithful]] Noetherian module over it, then the ring is a Noetherian ring.<ref>{{harvnb|Matsumura|1989|loc=Theorem 3.5.}}</ref><!-- not sure if “commutative” can be dropped. -->
* ([[Eakin–Nagata theorem|Eakin–Nagata]]) If a ring ''A'' is a [[subring]] of a commutative Noetherian ring ''B'' such that ''B'' is a [[finitely generated module]] over ''A'', then ''A'' is a Noetherian ring.<ref>{{harvnb|Matsumura|1989|loc=Theorem 3.6.}}</ref>
*Similarly, if a ring ''A'' is a subring of a commutative Noetherian ring ''B'' such that ''B'' is [[faithfully flat ring homomorphism|faithfully flat]] over ''A'' (or more generally exhibits ''A'' as a [[pure subring]]), then ''A'' is a Noetherian ring (see the "faithfully flat" article for the reasoning).
* Every [[Localization of a ring|localization]] of a commutative Noetherian ring is Noetherian.
* A consequence of the [[Hopkins–Levitzki theorem|Akizuki–Hopkins–Levitzki theorem]] is that every left [[Artinian ring]] is left Noetherian.  Another consequence is that a left Artinian ring is right Noetherian if and only if it is right Artinian.  The analogous statements with "right" and "left" interchanged are also true.
* A left Noetherian ring is left [[coherent ring|coherent]] and a left Noetherian [[Domain (ring theory)|domain]] is a left [[Ore domain]].
* (Bass) A ring is (left/right) Noetherian if and only if every [[direct sum of modules|direct sum]] of [[injective module|injective]] (left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of [[indecomposable module|indecomposable]] injective modules.<ref name="Bass injective">{{harvnb|Anderson|Fuller|1992|loc=Proposition 18.13.}}</ref> See also [[#Implication on injective modules]] below.
* In a commutative Noetherian ring, there are only finitely many [[minimal prime ideal]]s. Also, the [[descending chain condition]] holds on prime ideals.
* In a commutative Noetherian domain ''R'', every element can be factorized into [[irreducible element]]s (in short, ''R'' is a [[factorization domain]]). Thus, if, in addition, the factorization is unique [[up to]] multiplication of the factors by [[unit (ring theory)|unit]]s, then ''R'' is a [[unique factorization domain]].