Noetherian ring

A Noetherian ring is one where you can never keep making ideals strictly bigger forever — any attempt to build an endless increasing chain of ideals must eventually stall. This finiteness condition, named for Emmy Noether, is the single most important taming hypothesis in commutative algebra; almost every ring arising in geometry and number theory satisfies it.

Precisely, a ring R is (left) Noetherian if it satisfies the ascending chain condition on (left) ideals: every increasing chain I_1 ⊆ I_2 ⊆ I_3 ⊆ ... of ideals stabilizes, meaning I_n = I_{n+1} = ... from some point on. Two equivalent formulations are often more convenient: every ideal is finitely generated, and every nonempty collection of ideals has a maximal element with respect to inclusion.

The reason Noetherian rings are everywhere is Hilbert's basis theorem: if R is Noetherian then so is the polynomial ring R[x]. Combined with the fact that quotients of Noetherian rings are Noetherian, this guarantees that finitely generated algebras over a field or over Z are Noetherian, which underwrites all of classical algebraic geometry. Not every ring is Noetherian, though — the polynomial ring k[x_1, x_2, x_3, ...] in infinitely many variables fails the chain condition.