symmetric monoidal (∞,1)-category of spectra
(In this section, “ring” means rng, where the presence of a multiplicative identity is not assumed unless we say “unital ring”.)
A (left) noetherian ring is a ring for which every ascending chain of its (left) ideals stabilizes. In other words, it is noetherian if its underlying -module is a noetherian object in the category of left -modules (recall that a left ideal is simply a submodule of ). Similarly for right noetherian rings. Left noetherianness is independent of right noetherianness. A ring is noetherian if it is both left noetherian and right noetherian.
An equivalent condition is that all (left) ideals are finitely generated.
A dual condition is artinian: an artinian ring is a ring satisfying the descending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian; artinian rings are intuitively much smaller than generic noetherian rings.
Spectra of noetherian rings are glued together to define locally noetherian schemes.
One of the best-known properties is the Hilbert basis theorem. Let be a (unital) ring.
(Hilbert) If is left Noetherian, then so is the polynomial algebra . (Similarly if “right” is substituted for “left”.)
(We adapt the proof from Wikipedia.) Suppose is a left ideal of that is not finitely generated. Using the axiom of dependent choice, there is a sequence of polynomials such that the left ideals form a strictly increasing chain and is chosen to have degree as small as possible. Putting , we have . Let be the leading coefficient of . The left ideal of is finitely generated; say generates. Thus we may write
The polynomial belongs to , so belongs to . Also has degree or less, and therefore so does . But notice that the coefficient of in is zero, by (1). So in fact has degree less than , contradicting how was chosen.