Primary decomposition in Noetherian rings

Lasker–Noether theorem. Let $R$ be a commutative Noetherian ring and let $I\subsetneq R$ be a proper ideal. Then there exist primary ideals $Q_1,\dots,Q_r$ such that

I = Q_1 \cap \cdots \cap Q_r .

One may choose the decomposition to be minimal: no component is redundant and the radicals $\sqrt{Q_i}$ are distinct prime ideals. For every minimal decomposition, the set

\{\sqrt{Q_1},\dots,\sqrt{Q_r}\}

depends only on $I$ , although the primary components themselves need not be unique.

Primary ideals

An ideal $Q\subsetneq R$ is primary if $ab\in Q$ and $a\notin Q$ imply $b^n\in Q$ for some $n\geq 1$ . Equivalently, every zero divisor in $R/Q$ is nilpotent. See primary decomposition for the general terminology.

Examples

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).

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 $(x,y)$ .

In the integers. In $\mathbb Z$ , $(12) = (4) \cap (3).$ The ideal $(4)$ is $(2)$ -primary and $(3)$ is prime (hence primary).