Primary decomposition in Noetherian rings

Primary decomposition is a way to express an ideal as an intersection of “power-of-a-prime” pieces. The phenomenon is special to Noetherian rings and is formalized by the Lasker–Noether theorem . See also primary decomposition for the general language.

Definition (primary ideal)

Let $R$ be a commutative ring and let $Q \subsetneq R$ be an ideal.
$Q$ is primary if whenever $ab \in Q$ and $a \notin Q$, there exists $n \ge 1$ such that $b^n \in Q$.

Equivalently, in $R/Q$ every zero-divisor is nilpotent.

Theorem (Noetherian primary decomposition)

Let $R$ be a commutative Noetherian ring and let $I \subseteq R$ be an ideal. Then there exist primary ideals $Q_1,\dots,Q_r$ such that

One can choose the decomposition so that the radicals $\sqrt{Q_i}$ are distinct prime ideals . In any minimal primary decomposition (no redundant components and with distinct radicals), the set of prime ideals $\{\sqrt{Q_1},\dots,\sqrt{Q_r}\}$ depends only on $I$, not on the choice of decomposition.

Examples

1. A reduced principal ideal in a polynomial ring.
   In $k[x,y]$, the ideal $(xy)$ decomposes as

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

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

2. A decomposition with an embedded component.
   In $k[x,y]$,

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

   Indeed, if $f \in (x) \cap (x^2,y)$, then $f=xg$ and $xg \in (x^2,y)$ forces $g \in (x,y)$, so $f \in (x^2,xy)$.
   The ideal $(x)$ is prime, and $(x^2,y)$ is $(x,y)$-primary since its radical is $(x,y)$.

3. In the integers.
   In $\mathbb Z$,

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

   The ideal $(4)$ is $(2)$-primary and $(3)$ is prime (hence primary).