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# primary decomposition

Let $R$ be a commutative ring and $A$ be an ideal in $R$. A *primary 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.

## Mathematics Subject Classification

13C99*no label found*

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## Comments

## rad ?

What does 'rad(Qi)' mean? Is there a definition of 'rad'?

## Re: rad ?

Yup. It's the radical of the ideal. I'll make a link in the entry.

Cam