Editing Commutative ring (section)

=== Noetherian rings ===
{{Main|Noetherian ring}}
A ring is called ''Noetherian'' (in honor of [[Emmy Noether]], who developed this concept) if every [[ascending chain condition|ascending chain of ideals]]
<math display="block"> 0 \subseteq I_0 \subseteq I_1 \subseteq \dots \subseteq I_n \subseteq I_{n+1} \dots </math>
becomes stationary, i.e. becomes constant beyond some index <math> n </math>. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, [[submodule]]s of finitely generated modules are finitely generated.

Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry. For example, if <math> R </math> is Noetherian, then so is the polynomial ring <math> R \left[X_1,X_2,\dots,X_n\right] </math> (by [[Hilbert's basis theorem]]), any localization <math> S^{-1}R </math>, and also any factor ring <math> R / I </math>.

Any non-Noetherian ring <math> R </math> is the [[union (set theory)|union]] of its Noetherian subrings. This fact, known as [[Noetherian approximation]], allows the extension of certain theorems to non-Noetherian rings.