primary decomposition
PrimaryDecomposition
2004-03-12 13:14:10
2008-10-05 11:31:34
Definition
decomposable ideal
minimal primary decomposition
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Let $R$ be a commutative ring and $A$ be an ideal in $R$. A \emph{\PMlinkescapetext{primary} decomposition} of $A$ is a way of writing $A$ as a finite intersection of primary ideals:
\begin{align*}
A=\bigcap_{i=1}^n Q_i,
\end{align*}
where the $Q_i$ are primary in $R$.
Not every ideal admits a primary decomposition, so we define a \emph{decomposable ideal} to be one that does.
\textbf{Example}. Let $R=\Z$ and take $A=(180)$. Then $A$ is decomposable, and a primary decomposition of $A$ is given by
\begin{align*}
A=(4)\cap (9)\cap (5),
\end{align*}
since $(4)$, $(9)$, and $(5)$ are all primary ideals in $\Z$.
Given a primary decomposition $A=\cap Q_i$, we say that the decomposition is a \emph{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
\begin{align*}
Q_i\not\subset \bigcap_{j\neq i} Q_j
\end{align*}
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.