Primary decomposition

Let $R$ be a commutative ring and $I\subseteq R$ an ideal .

Definition

A primary decomposition of $I$ is an expression

where each $Q_i$ is a primary ideal . Typically one tracks the associated prime ideals

where $\sqrt{Q_i}$ is the radical of $Q_i$ (and each $P_i$ is a prime ideal ).

Theorem (existence in the Noetherian case)

If $R$ is a Noetherian ring , then every ideal $I$ admits a primary decomposition. This is the content of the Lasker–Noether theorem (see also Noetherian primary decomposition ).

Examples

1. A squarefree monomial ideal.
   In $k[x,y]$,

   $$(xy) \;=\; (x)\cap (y).$$

   Here $(x)$ and $(y)$ are prime (hence primary).

2. A slightly less trivial example.
   In $k[x,y]$,

   $$(x^2,xy) \;=\; (x)\cap (x^2,y).$$

   The ideal $(x)$ is prime, and $(x^2,y)$ is $(x,y)$-primary.

3. An integer ideal.
   In $\mathbb{Z}$,

   $$(12) \;=\; (4)\cap (3).$$

   Indeed $(4)\cap (3)=(\mathrm{lcm}(4,3))=(12)$; moreover $(4)=(2^2)$ is $(2)$-primary and $(3)$ is prime.