# primary decomposition

Let $R$ be a commutative ring and $A$ be an ideal in $R$. A decomposition of $A$ is a way of writing $A$ as a finite intersection of primary ideals:

 $\displaystyle A=\bigcap_{i=1}^{n}Q_{i},$

where the $Q_{i}$ are primary in $R$.

Not every ideal admits a primary decomposition, so we define a decomposable ideal to be one that does.

Example. Let $R=\mathbb{Z}$ and take $A=(180)$. Then $A$ is decomposable, and a primary decomposition of $A$ is given by

 $\displaystyle A=(4)\cap(9)\cap(5),$

since $(4)$, $(9)$, and $(5)$ are all primary ideals in $\mathbb{Z}$.

Given a primary decomposition $A=\cap Q_{i}$, we say that the decomposition is a minimal primary decomposition if for all $i$, the prime ideals $P_{i}=\text{rad}(Q_{i})$ (where rad denotes the radical of an ideal) are distinct, and for all $1\leq i\leq n$, we have

 $\displaystyle Q_{i}\not\subset\bigcap_{j\neq i}Q_{j}$

In the example above, the decomposition $(4)\cap(9)\cap(5)$ of $A$ is minimal, where as $A=(2)\cap(4)\cap(3)\cap(9)\cap(5)$ is not.

Every primary decomposition can be refined to admit a minimal primary decomposition.

Title primary decomposition PrimaryDecomposition 2013-03-22 14:15:05 2013-03-22 14:15:05 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 13C99 shortest primary decomposition decomposable ideal minimal primary decomposition