Noetherian

Noetherian is a term used in abstract algebra, named after Emmy Noether. A ring is called Noetherian if every ideal is finitely generated, which is equivalent to the ascending chain condition: every increasing sequence of ideals eventually stabilizes. The notion extends to modules: a module over a ring is Noetherian if every submodule is finitely generated, equivalently if every ascending chain of submodules stabilizes.

For rings, being Noetherian has several important consequences. Quotients of Noetherian rings are Noetherian, and if a ring is Noetherian then so are its polynomial rings in finitely many indeterminates. More generally, a finitely generated algebra over a Noetherian ring is Noetherian. A module is Noetherian if it is finitely generated over a Noetherian ring, and every submodule of such a module is finitely generated.

Examples and non-examples help illustrate the concept. The ring of integers Z is Noetherian, and every principal ideal domain is Noetherian. The polynomial ring k[x1, ..., xn] over a field k is Noetherian for any finite n, by the Hilbert basis theorem; but a polynomial ring in infinitely many variables, such as k[x1, x2, x3, ...], is not Noetherian. Z[x] is Noetherian, while Z[x1, x2, ...] is not.

Noetherian rings possess favorable properties for geometry and computation. Localizations and finite products of Noetherian rings are Noetherian. Artinian rings are Noetherian, but the converse fails in general (for example, Z is Noetherian but not Artinian). Noetherian conditions underpin many results in algebraic geometry, as coordinate rings of affine varieties are Noetherian.