Primary Decomposition - Main Theorem

The existence and uniqueness of primary decomposition in Noetherian rings form one of the foundational results of commutative algebra.

TheoremLasker-Noether Theorem (Existence)

Let $R$ be a Noetherian ring and $I \subsetneq R$ a proper ideal. Then $I$ admits a primary decomposition: $I = Q_1 \cap Q_2 \cap \cdots \cap Q_n$ where each $Q_i$ is primary. Moreover, $I$ has a minimal primary decomposition.

The Noetherian hypothesis is essential. In non-Noetherian rings, ideals may fail to have primary decompositions. The prototypical example is $k[x_1, x_2, x_3, \ldots]$ where the ideal $(x_1, x_2^2, x_3^3, \ldots)$ cannot be expressed as a finite intersection of primary ideals.

TheoremUniqueness of Associated Primes

If $I = Q_1 \cap \cdots \cap Q_n$ is a minimal primary decomposition with $\mathfrak{p}_i = \sqrt{Q_i}$ , then the set $\{\mathfrak{p}_1, \ldots, \mathfrak{p}_n\}$ is uniquely determined by $I$ .

These primes are called the associated primes of $I$ , denoted $\text{Ass}(R/I)$ . They can be characterized without reference to primary decomposition as: $\text{Ass}(R/I) = \{\mathfrak{p} : \mathfrak{p} = \text{ann}(x + I) \text{ for some } x \in R\}$

ExampleComputing Associated Primes

For $I = (x^2, xy) \subset k[x,y]$ :

$\text{ann}(\overline{y}) = (x)$ is prime
$\text{ann}(\overline{1}) = I$ with $\sqrt{I} = (x)$
Associated primes: $\{(x), (x,y)\}$

The prime $(x)$ is isolated (minimal), while $(x,y)$ is embedded.

TheoremFirst Uniqueness Theorem

In a minimal primary decomposition $I = Q_1 \cap \cdots \cap Q_n$ :

The isolated (minimal) primary components $Q_i$ corresponding to minimal associated primes are uniquely determined
The embedded primary components corresponding to non-minimal associated primes may not be unique
However, all primary decompositions have the same associated primes

Remark

The height of a prime $\mathfrak{p}$ is the maximal length of chains of prime ideals descending from $\mathfrak{p}$ . Embedded primes always have height strictly greater than some associated minimal prime, reflecting their "lower-dimensional" geometric nature.

TheoremCharacterization via Localization

A prime $\mathfrak{p}$ is associated to $I$ if and only if $I R_\mathfrak{p}$ is $\mathfrak{p}$ -primary in $R_\mathfrak{p}$ (equivalently, $\mathfrak{p} R_\mathfrak{p}$ -primary).

This local characterization allows computation of associated primes by localizing and checking primality conditions locally.

TheoremFinite Set of Associated Primes

In a Noetherian ring, every ideal has only finitely many associated primes. This follows from the Noetherian property ensuring finite primary decompositions.

The minimal associated primes are precisely the minimal primes over $I$ , connecting associated primes to the geometric notion of irreducible components.

ExampleApplication to Zero Divisors

The set of zero divisors on $R/I$ equals: $\bigcup_{\mathfrak{p} \in \text{Ass}(R/I)} \mathfrak{p}$

This characterizes zero divisors as precisely those elements belonging to some associated prime, providing a clean algebraic description.

These theorems establish primary decomposition as a fundamental tool, generalizing unique factorization to ideals while respecting geometric intuition about irreducible components.