Primary decomposition in Noetherian rings

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

Writing $\mathfrak p_i=\sqrt{Q_i}$ (the radical ), each $\mathfrak p_i$ is a prime ideal . One can choose a minimal primary decomposition in which the primes $\mathfrak p_i$ are distinct; in that case the set $\{\mathfrak p_i\}$ is uniquely determined by $I$.

This is commonly packaged as the Lasker–Noether theorem , together with the definition of primary decomposition .

Related knowls.

• Primary decomposition
• Lasker–Noether theorem
• Primary ideal

Examples

1. In $\mathbb{Z}$: factorization into primary components.
   In $R=\mathbb{Z}$,

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

   Here $(4)$ is $(2)$-primary and $(3)$ is $(3)$-primary. (In a PID, $(a)\cap(b)=(\mathrm{lcm}(a,b))$.)

2. A squarefree monomial ideal.
   In $k[x,y]$,

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

   Both $(x)$ and $(y)$ are prime (hence primary), so this is a primary decomposition.

3. A nontrivial primary decomposition with the same radical.
   In $k[x,y]$,

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

   The ideals $(x^2,y)$ and $(x,y^2)$ are both $(x,y)$-primary (their radicals are $(x,y)$), and their intersection gives $(x,y)^2=(x^2,xy,y^2)$.