Lasker–Noether theorem

In a commutative ring $A$, an ideal $Q$ is primary if $ab\in Q$ implies $a\in Q$ or $b^n\in Q$ for some $n\ge 1$; equivalently, the zero-divisors in $A/Q$ are nilpotent. If $Q$ is primary and $\sqrt{Q}=\mathfrak p$ is prime, then $Q$ is called $\mathfrak p$-primary. A representation of an ideal as an intersection of primary ideals is a primary decomposition.

Theorem (Lasker–Noether). Let $A$ be a Noetherian ring and let $I\subseteq A$ be an ideal. Then there exist finitely many primary ideals $Q_1,\dots,Q_r\subseteq A$ such that

Moreover, one can choose the decomposition so that the radicals $\mathfrak p_i=\sqrt{Q_i}$ are distinct prime ideals; in such a minimal primary decomposition, the set of primes $\{\mathfrak p_i\}$ is uniquely determined by $I$ (it is the set of prime ideals minimal over $I$).

Via the Zariski topology on the prime spectrum $\operatorname{Spec}(A)$, the identity $I=\bigcap Q_i$ translates to the geometric union

so primary decomposition packages the "irreducible pieces" of $V(I)$.

Examples

1. Integers: prime-power pieces. In $A=\mathbb Z$, every ideal is principal. For $I=(12)$ one has
   $$(12)=(4)\cap (3).$$
   Here $(4)$ is $(2)$-primary and $(3)$ is $(3)$-primary.

1. A union of coordinate axes. In $A=k[x,y]$ (with $k$ a field), the ideal $I=(xy)$ decomposes as
   $$(xy)=(x)\cap (y).$$
   Both $(x)$ and $(y)$ are prime (hence primary), corresponding to the two axes in $V(xy)$.

1. A primary component with embedded nilpotents. In $A=k[x,y]$, the ideal $I=(x^2,xy)$ admits the decomposition
   $$(x^2,xy)=(x)\cap (x^2,y).$$
   The ideal $(x)$ is prime, while $(x^2,y)$ is $(x,y)$-primary since its radical is $(x,y)$ and $x$ becomes nilpotent modulo $(x^2,y)$.