Noetherian Rings and Modules - Examples and Constructions

Constructing and recognizing Noetherian rings requires understanding how the property behaves under various operations.

TheoremHilbert's Basis Theorem

If $R$ is Noetherian, then the polynomial ring $R[x]$ is Noetherian.

By induction, $R[x_1, \ldots, x_n]$ is Noetherian. This is one of the most important results in commutative algebra, ensuring finitely generated algebras over fields are Noetherian.

ExampleApplications of Hilbert's Basis Theorem

$k[x_1, \ldots, x_n]$ is Noetherian for any field $k$
Every finitely generated $k$ -algebra is Noetherian (as a quotient of polynomial ring)
Coordinate rings of affine varieties are Noetherian
Ideal theory in polynomial rings has finiteness properties

Without this theorem, algebraic geometry would lose most of its finiteness results.

ExamplePower Series Rings

If $R$ is Noetherian, then $R[[x]]$ (formal power series) is Noetherian. More generally, $R[[x_1, \ldots, x_n]]$ is Noetherian.

This follows from a variant of Hilbert's Basis Theorem. Complete local rings are Noetherian when their residue fields are.

DefinitionGraded Rings and Modules

A ring $R = \bigoplus_{n \geq 0} R_n$ is graded if $R_i R_j \subseteq R_{i+j}$ . A graded $R$ -module $M = \bigoplus_{n \in \mathbb{Z}} M_n$ satisfies $R_i M_j \subseteq M_{i+j}$ .

Many geometric objects (projective varieties, sheaves) naturally carry graded structures.

TheoremGraded Version of Hilbert's Basis Theorem

If $R_0$ is Noetherian and $R$ is finitely generated as an $R_0$ -algebra, then $R$ is Noetherian.

For instance, if $R = R_0[x_1, \ldots, x_n]$ with $\deg(x_i) > 0$ , then $R$ is Noetherian.

ExampleHomogeneous Coordinate Rings

The homogeneous coordinate ring of projective space $\mathbb{P}^n$ over field $k$ is: $S = k[x_0, \ldots, x_n]$

graded with $\deg(x_i) = 1$ . This is Noetherian by Hilbert's Basis Theorem. The Proj construction uses this graded ring to build $\mathbb{P}^n$ geometrically.

DefinitionCohen's Structure Theorem

Complete Noetherian local rings $(R, \mathfrak{m}, k)$ with residue field $k$ containing a field have particularly nice structures. Cohen's Structure Theorem states such rings admit surjections from power series rings: $k[[x_1, \ldots, x_n]] \to R$

This provides "coordinates" for complete local rings, fundamental in deformation theory.

ExampleNon-Noetherian Completions

While completions of Noetherian local rings are Noetherian, completions in non-Noetherian settings can fail to be Noetherian. The Nagata criterion characterizes when completions remain Noetherian.

For instance, certain valuation rings have non-Noetherian completions despite being Noetherian themselves.

TheoremEakin-Nagata Theorem

If $R \subseteq S$ with $S$ finitely generated as an $R$ -module and $S$ Noetherian, then $R$ is Noetherian.

This surprising result shows the Noetherian property can "descend" under finite extensions, unlike most ring properties.

ExampleCoordinate Rings of Varieties

For an affine variety $V \subseteq \mathbb{A}^n$ over an algebraically closed field $k$ , the coordinate ring: $k[V] = k[x_1, \ldots, x_n]/I(V)$

is Noetherian (quotient of Noetherian ring). Every function on $V$ can be expressed using finitely many generators, reflecting geometric finiteness.

DefinitionExcellent Rings

A Noetherian ring is excellent if:

It is universally catenary
Formal fibers are geometrically regular
Regular loci are open in finite type algebras

Excellent rings (introduced by Grothendieck) include most "naturally occurring" Noetherian rings: fields, $\mathbb{Z}$ , complete local rings, and finitely generated algebras over these. They enjoy strong geometric properties.

ExampleRings of Integers

For a number field $K$ , the ring of integers $\mathcal{O}_K$ is:

Noetherian (finitely generated as $\mathbb{Z}$ -module)
Dedekind (one-dimensional, integrally closed)
Excellent

These properties make $\mathcal{O}_K$ amenable to both algebraic and geometric techniques.

Remark

The class of Noetherian rings is closed under:

Quotients
Localizations
Polynomial extensions (Hilbert's Basis Theorem)
Finite extensions (module-finite)
Completions (for local rings)

This closure makes Noetherian rings ubiquitous and ensures the property persists through standard constructions.

These examples and constructions demonstrate the robustness and universality of the Noetherian condition across algebra and geometry.