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date: 2022-01-24 14:16
modification date: Monday 24th January 2022 14:16:07
title: Noetherian
aliases: [Noetherian ring]

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- Tags: 
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- Refs:
	- #todo/add-references
- Links:
	- [Commutative Algebra](Projects/2022%20Advanced%20Qual%20Projects/Commutative%20Algebra/000%20Resources.md)

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# Noetherian


# Noetherian Rings

Satisfy the ACC on ideals.

# Examples

![](attachments/Pasted%20image%2020220420181700.png)

- $\OO_{X, x}$ for $\OO_X(\wait) \da \Hol(\wait; \CC)$ the sheaf of holomorphic functions, the stalks are rings of germs of holomorphic functions. These are regular local rings, and Noetherian by the [Weierstrass preparation theorem](Weierstrass%20preparation%20theorem). 

## Non-Noetherian Rings

- $R[x_0, x_1, \cdots]$, a polynomial ring on countably many variables.
- For $X = \CC$ equipped with the sheaf  $\mcf = C^\infty(\wait; \RR)$, the stalk $\mcf_0$ is the ring of germs of smooth functions at the origin. Take $f(x) \da e^{-{1\over x^2}}$ and patch $f(0) = 0$ to get a smooth-nonanaltic function, contradicting [Krull's intersection theorem](Unsorted/Krull's%20intersection%20theorem.md).
- $C^0([0, 1]; \RR)$ the ring of continuous functions.
- $\OO_K$ for $K = \CCpadic = \widehat{\algcl(\QQ_p)}$

# Exercises

- Show that [Noetherian](Noetherian.md) [integral domains](integral%20domains) have factorization into finitely many irreducibles.
- Show that subrings of [Noetherian](Noetherian.md) rings need not be Notherian.
- Show that subalgebras of [finitely generated](finitely%20generated.md) [algebras](algebras) need not be finitely generated.
- Show that if $R$ is Noetherian, then the [nilradical](nilradical) $\nilrad{R}^n = 0$ for some $n$.
- Show that the [Krull's intersection theorem](Krull's%20intersection%20theorem.md) fails for non-Noetherian rings.
- Find a [Noetherian](Noetherian.md) ring where each ideal is [finitely generated](finitely%20generated.md), but the number of generators needed is not uniformly bounded.
- Show that in a [Noetherian](Noetherian.md) ring, the only element in the intersection of all powers of all maximal ideals is zero.
- Show that a subring of a [Noetherian](Noetherian.md) ring need not be Noetherian.
- Does [Noetherian](Noetherian.md) imply [Artinian](Artinian)? Or vice-versa?
	- What is a sufficient condition to guarantee [Noetherian](Noetherian.md) $\iff$ [Artinian](Artinian)?
- Prove several equivalent characterizations of [Noetherian](Noetherian.md) rings.
- Show that the homomorphic image of a [Noetherian](Noetherian.md) ring is Noetherian.
- If $R$ is a Noetherian ring, show that any [finitely generated](finitely%20generated.md) module $M\in \rmod^\fg$ is Noetherian.
- Show that if $R$ is [Noetherian](Noetherian.md), any [Unsorted/localization of rings](Unsorted/localization%20of%20rings.md) is Noetherian.
- Show that if $R$ is [Noetherian](Noetherian.md) then $R[x]$ is Noetherian.
- Show that every $A\in \Alg^\fg\slice k$ for $k\in \Field$ is [Noetherian](Noetherian.md).
- Show that if $A \in \Alg^\fg\slice k$ for $k\in \Field$, then if $A$ is additionally a field then $A\slice k$ is a finite algebraic extension of fields.
- Show that $\nilrad{R}$ is nilpotent when $R$ is Noetherian.
- Show that localizations preserve exactness and being Noetherian.
- Show that the $\mfm\dash$adic completion of a Noetherian [local ring](local%20ring) is again local.
- Show that every ideal in a Noetherian ring has finite [height](height.md), but may not be finite [depth](depth.md).
	- Show that ideals in a local Noetherian ring are finite [depth](depth.md).
- Show that Noetherian rings are finite dimensional.
- Show that a [graded ring](graded%20ring) $R$ is Noetherian iff $R_0$ is Noetherian and $R$ is a [finitely generated](finitely%20generated.md) $R_0\dash$algebra.