Noetherian Rings and Modules - Applications

The Noetherian condition enables powerful applications in dimension theory, primary decomposition, and algebraic geometry.

TheoremFinite Associated Primes

In a Noetherian ring $R$ , every ideal $I$ has only finitely many associated primes. More generally, every finitely generated module $M$ over a Noetherian ring has finitely many associated primes.

This finiteness is essential for primary decomposition and understanding zero divisors.

ExampleNon-Noetherian Counterexample

In $R = k[x_1, x_2, x_3, \ldots]/(x_1^2, x_2^2, x_3^2, \ldots)$ , the ideal $(0)$ has infinitely many associated primes: each $(x_i)$ is associated. This pathology cannot occur in Noetherian rings.

TheoremDimension and Noetherian Property

For a Noetherian ring $R$ , the Krull dimension is the supremum of lengths of chains of prime ideals: $\dim(R) = \sup\{n : \mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n\}$

The Noetherian property ensures this supremum is well-defined and finite for many natural rings, though Noetherian rings can have infinite dimension.

ExampleComputing Dimensions

$\dim(k[x_1, \ldots, x_n]) = n$ : polynomial rings
$\dim(\mathbb{Z}) = 1$ : only chains $(0) \subsetneq (p)$
$\dim(k[[x_1, \ldots, x_n]]) = n$ : power series rings
$\dim(k[x_1, \ldots, x_n]/I) =$ dimension of variety $V(I)$

TheoremCatenary Rings

A Noetherian ring $R$ is catenary if for every pair of prime ideals $\mathfrak{p} \subseteq \mathfrak{q}$ , all maximal chains of primes between them have the same length.

Cohen-Macaulay rings and most geometric rings are catenary, ensuring dimension behaves predictably.

Remark

Universally catenary rings remain catenary after any finite type extension. These include complete local rings and finitely generated algebras over fields, forming the backbone of excellent ring theory.

TheoremArtinian Rings and Length

A Noetherian ring $R$ is Artinian if and only if $\dim(R) = 0$ . In this case, $R$ has finite length as an $R$ -module, and: $\ell(R) = \sum_{\mathfrak{m} \text{ maximal}} \ell(R_\mathfrak{m})$

Artinian rings are "zero-dimensional" and completely understood via composition series.

ExampleStructure of Artinian Rings

Every Artinian ring decomposes (by Chinese Remainder Theorem) as: $R \cong R_1 \times \cdots \times R_n$ where each $R_i$ is a local Artinian ring. Local Artinian rings have the form $(R, \mathfrak{m})$ with $\mathfrak{m}^k = 0$ for some $k$ , making them nilpotent-thickened fields.

TheoremHilbert-Samuel Polynomial

For a Noetherian local ring $(R, \mathfrak{m})$ and finitely generated $R$ -module $M$ , the function: $n \mapsto \ell(M/\mathfrak{m}^n M)$ agrees with a polynomial in $n$ for large $n$ , called the Hilbert-Samuel polynomial.

The degree of this polynomial equals $\dim(M)$ , and its leading coefficient encodes the multiplicity of $M$ .

ExampleHilbert Functions

For $R = k[x_1, \ldots, x_d]$ graded and $M$ finitely generated graded, the Hilbert function $H_M(n) = \dim_k(M_n)$ is eventually polynomial in $n$ .

For $R = k[x,y]$ and $M = R$ , we have $H_M(n) = \binom{n+1}{1} = n+1$ , a polynomial of degree $\dim(R) = 2$ minus 1.

TheoremNoetherian Schemes

A scheme $X$ is Noetherian if it is covered by finitely many open affine subschemes $\text{Spec}(R_i)$ with each $R_i$ Noetherian.

Noetherian schemes satisfy:

Every open has finitely many irreducible components
Every ascending chain of closed subschemes stabilizes
Quasi-coherent sheaves are generated by global sections locally

These properties make Noetherian schemes geometrically well-behaved.

Remark

Most schemes in algebraic geometry are Noetherian: varieties over fields, schemes over $\text{Spec}(\mathbb{Z})$ , and completions of these. Non-Noetherian schemes arise in infinite-dimensional contexts like loop groups or ind-schemes.

These applications demonstrate how the Noetherian condition enables dimension theory, primary decomposition, and finite presentations throughout commutative algebra and geometry.