Lasker–Noether theorem

Lasker–Noether theorem. Let $A$ be a Noetherian commutative ring, and let $I\subseteq A$ be an ideal. Then there are primary ideals $Q_1,\dots,Q_r\subseteq A$ such that

I=\bigcap_{i=1}^r Q_i.

The decomposition may be chosen irredundant with distinct radicals $\mathfrak p_i=\sqrt{Q_i}$ . In any such minimal primary decomposition, the set $\{\mathfrak p_i\}$ is uniquely determined by $I$ ; it is the set of associated primes of $A/I$ . Its inclusion-minimal members are precisely the prime ideals minimal over $I$ .

Here an ideal $Q$ is primary if $ab\in Q$ implies $a\in Q$ or $b^n\in Q$ for some $n\ge 1$ . Such an expression for $I$ is a primary decomposition.

Geometric interpretation

In the Zariski topology on the prime spectrum $\operatorname{Spec}(A)$ , the decomposition gives

V(I)=\bigcup_{i=1}^r V(Q_i),

with $V(Q_i)=V(\mathfrak p_i)$ . Embedded associated primes contribute scheme-theoretic information but do not give additional irreducible components of the underlying closed set.

Examples

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

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, corresponding to the two axes in $V(xy)$ .

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. The latter is an embedded component.