# nLab noetherian ring

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A Noetherian (or often, as below, noetherian) ring (or rng) is one where it is possible to do induction over its ideals, because the ordering of ideals by reverse inclusion is well-founded.

## Definition

(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 $R$ 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.

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.

## Properties

One of the best-known properties is the Hilbert basis theorem. Let $R$ be a (unital) ring.

###### Theorem

(Hilbert) If $R$ is left Noetherian, then so is the polynomial algebra $R[x]$. (Similarly if “right” is substituted for “left”.)

###### Proof

(We adapt the proof from Wikipedia.) Suppose $I$ is a left ideal of $R[x]$ that is not finitely generated. Using the axiom of dependent choice, there is a sequence of polynomials $f_n \in I$ such that the left ideals $I_n \coloneqq (f_0, \ldots, f_{n-1})$ form a strictly increasing chain and $f_n \in I \setminus I_n$ is chosen to have degree as small as possible. Putting $d_n \coloneqq \deg(f_n)$, we have $d_0 \leq d_1 \leq \ldots$. Let $a_n$ be the leading coefficient of $f_n$. The left ideal $(a_0, a_1, \ldots)$ of $R$ is finitely generated; say $(a_0, \ldots, a_{k-1})$ generates. Thus we may write

(1)$a_k = \sum_{i=0}^{k-1} r_i a_i$

The polynomial $g = \sum_{i=0}^{k-1} r_i x^{d_k - d_i} f_i$ belongs to $I_k$, so $f_k - g$ belongs to $I \setminus I_k$. Also $g$ has degree $d_k$ or less, and therefore so does $f_k - g$. But notice that the coefficient of $x^{d_k}$ in $f_k - g$ is zero, by (1). So in fact $f_k - g$ has degree less than $d_k$, contradicting how $f_k$ was chosen.

## References

Revised on July 4, 2015 08:00:07 by Todd Trimble (67.81.95.215)