Primary decomposition in Noetherian rings - Imported Knowlpedia Content

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

I = Q_1 \cap \cdots \cap Q_r .

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

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).