Commutative Ring - Noetherian Rings

Noetherian Rings

A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals

0 ⊆ I₀ ⊆ I₁ ... ⊆ I_n ⊆ I_{n + 1} ⊆ ...

becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. A ring is called Artinian (after Emil Artin), if every descending chain of ideals

R ⊇ I₀ ⊇ I₁ ... ⊇ I_n ⊇ I_{n + 1} ⊇ ...

becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, Z is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain

Z ⊋ 2Z ⊋ 4Z ⊋ 8Z ⊋ ...

shows. In fact, by the Hopkins–Levitzki theorem, every Artinian ring is Noetherian.

Being Noetherian is an extremely important finiteness condition. The condition is preserved under many operations that occur frequently in geometry: if R is Noetherian, then so is the polynomial ring R (by Hilbert's basis theorem), any localization S−1R, factor rings R / I.