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:

$A={\displaystyle \bigcap _{i=1}^n}{Q}_i,$

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

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

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 $1\le i\le n$, we have

$Q_i\not\subset {\displaystyle \bigcap _{j\ne i}}{Q}_j$

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.