symmetric monoidal (∞,1)-category of spectra
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.
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.