nLab
noetherian ring

Contents

Definition

A (left) noetherian ring RR is a ring for which every ascending chain of its (left) ideals stabilizes. In other words, it is noetherian if its underlying RR-module RR{}_R R is a noetherian object in the category RModR Mod of left RR-modules (recall that a left ideal is simply a submodule of RR{}_R R). 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.

References

Revised on December 4, 2013 03:47:57 by Urs Schreiber (82.113.99.55)