# nLab noetherian ring

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A (left) noetherian ring $R$ is a ring for which every ascending chain of its (left) ideals stabilizes. In other words, it is noetherian if its underlying $R$-module ${}_R R$ is a noetherian object in the category $R Mod$ of left $R$-modules (recall that a left ideal is simply a submodule of ${}_R R$). 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 August 18, 2014 18:21:54 by Urs Schreiber (89.204.135.46)