# 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.

Title | primary decomposition |
---|---|

Canonical name | PrimaryDecomposition |

Date of creation | 2013-03-22 14:15:05 |

Last modified on | 2013-03-22 14:15:05 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13C99 |

Synonym | shortest primary decomposition |

Defines | decomposable ideal |

Defines | minimal primary decomposition |