Noetherian Rings and Modules - Key Properties

Noetherian rings enjoy remarkable stability properties under standard ring-theoretic operations, making them a robust class.

TheoremClosure Under Quotients

If $R$ is Noetherian and $I$ is an ideal, then $R/I$ is Noetherian.

Proof: Ideals of $R/I$ correspond bijectively to ideals of $R$ containing $I$ via the correspondence theorem. Since every ideal containing $I$ in $R$ is finitely generated (as $R$ is Noetherian), its image in $R/I$ is finitely generated.

TheoremLocalization Preserves Noetherian

If $R$ is Noetherian and $S \subseteq R$ is a multiplicative set, then $S^{-1}R$ is Noetherian.

Moreover, if $M$ is a Noetherian $R$ -module, then $S^{-1}M$ is a Noetherian $S^{-1}R$ -module.

This allows checking Noetherian properties locally, since $R$ is Noetherian if and only if $R_\mathfrak{m}$ is Noetherian for all maximal ideals $\mathfrak{m}$ .

ExampleLocal Rings

For a Noetherian ring $R$ and prime $\mathfrak{p}$ , the localization $R_\mathfrak{p}$ is a Noetherian local ring. This is fundamental in studying local properties geometrically.

For instance, $\mathbb{Z}_{(p)}$ is Noetherian with maximal ideal $(p)$ .

TheoremFinite Generation of Modules

Over a Noetherian ring $R$ , every finitely generated module $M$ is Noetherian.

Proof: By induction on the number of generators. For one generator, $M \cong R/I$ is Noetherian. For $n$ generators, an exact sequence $0 \to K \to R^n \to M \to 0$ with $K, R^n$ Noetherian implies $M$ is Noetherian (submodules and quotients of Noetherian modules are Noetherian).

Remark

The converse fails: a Noetherian module over a non-Noetherian ring need not have all submodules finitely generated globally, though it does by definition.

TheoremPrimary Decomposition Exists

In a Noetherian ring, every proper ideal admits a primary decomposition. This is the Lasker-Noether theorem, proved earlier.

The Noetherian hypothesis is essential—in non-Noetherian rings, primary decomposition may fail to exist.

ExampleAssociated Primes

For a Noetherian ring $R$ and ideal $I$ , the set $\text{Ass}(R/I)$ of associated primes is finite. This fails in non-Noetherian rings where infinitely many primes may be associated.

For $I = (x^2, xy) \subset k[x,y]$ , we have $\text{Ass}(k[x,y]/I) = \{(x), (x,y)\}$ , a finite set.

TheoremNoetherian Induction

In a Noetherian ring or module, every non-empty set of submodules has a maximal element. This allows proof by Noetherian induction: to prove a property $P$ holds for all submodules, assume it fails for some, take a maximal counterexample, and derive a contradiction.

This principle is used throughout commutative algebra for existence proofs.

ExampleUsing Noetherian Induction

To prove every ideal in a Noetherian ring has a primary decomposition, assume there's an ideal $I$ without one. Among all such ideals, choose a maximal one. Show this leads to contradiction by decomposing $I$ using its non-primality, creating larger ideals that must have decompositions.

DefinitionLength of Module

For a module $M$ , a composition series is a chain: $0 = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n = M$ where each $M_i/M_{i-1}$ is simple (has no proper non-zero submodules). The length $\ell(M)$ is the length of such a series, if it exists.

Artinian and Noetherian modules of finite length have well-defined lengths independent of the composition series chosen.

TheoremHopkins-Levitzki Theorem

A Noetherian ring $R$ is Artinian if and only if every prime ideal is maximal (equivalently, $\dim(R) = 0$ ).

Artinian rings are precisely the zero-dimensional Noetherian rings, having finite length as modules over themselves.

ExampleArtinian Rings

$k$ (fields): dimension 0, finitely many ideals
$\mathbb{Z}/n\mathbb{Z}$ : zero-dimensional, Artinian
Local Artinian rings $(R, \mathfrak{m})$ with $\mathfrak{m}^n = 0$ for some $n$

These appear in deformation theory and as completed local rings modulo powers of the maximal ideal.

These properties make Noetherian rings the natural setting for most of commutative algebra and algebraic geometry, providing finiteness without excessive restrictions.