# prime ideal factorization is unique

The following theorem^{} shows that the decomposition of an (integral) invertible ideal into its prime factors^{} is unique, if it exists. This applies to the ring of integers^{} in a number field or, more generally, to any Dedekind domain^{}, in which every nonzero ideal is invertible.

###### Theorem.

Let $I$ be an invertible ideal in an integral domain $R$, and that

$$I={\U0001d52d}_{1}{\U0001d52d}_{2}\mathrm{\cdots}{\U0001d52d}_{m}={\U0001d52e}_{1}{\U0001d52e}_{2}\mathrm{\cdots}{\U0001d52e}_{n}$$ |

are two factorizations of $I$ into a product^{} of prime ideals^{}. Then $m\mathrm{=}n$ and, up to reordering of the factors, ${\mathrm{p}}_{k}\mathrm{=}{\mathrm{q}}_{k}$ ($k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots}\mathrm{,}n$).

Here we allow the case where $m$ or $n$ is zero, in which case such an empty product is taken to be the full ring $R$.

###### Proof.

We use induction^{} on $m+n$. First, the case with $m+n=0$ is trivial, so suppose that $m+n>0$.
As the set of prime ideals ${\U0001d52d}_{k}$, ${\U0001d52e}_{k}$ is partially ordered by inclusion, there must be a minimal element. After reordering, without loss of generality we may suppose that it is ${\U0001d52d}_{1}$. Then

$${\U0001d52e}_{1}{\U0001d52e}_{2}\mathrm{\cdots}{\U0001d52e}_{n}\subseteq {\U0001d52d}_{1},$$ |

so $n\ge 1$. Furthermore, as ${\U0001d52d}_{1}$ is prime, this implies that ${\U0001d52e}_{k}\subseteq {\U0001d52d}_{1}$ for some $k$. After reordering the factors, we can take $k=1$, so that ${\U0001d52e}_{1}\subseteq {\U0001d52d}_{1}$.

As ${\U0001d52d}_{1}$ is minimal^{} among the prime factors, we have ${\U0001d52e}_{1}={\U0001d52d}_{1}$. Also, ${\U0001d52d}_{1}$ is a factor of the invertible ideal $I$ and so is itself invertible. Therefore, it can be cancelled from the products,

$${\U0001d52d}_{2}\mathrm{\cdots}{\U0001d52d}_{m}={\U0001d52e}_{2}\mathrm{\cdots}{\U0001d52e}_{n}.$$ |

The induction hypothesis gives $m=n$ and, after reordering, ${\U0001d52d}_{k}={\U0001d52e}_{k}$ for $k=2,\mathrm{\dots},n$. ∎

Title | prime ideal factorization is unique |
---|---|

Canonical name | PrimeIdealFactorizationIsUnique |

Date of creation | 2013-03-22 18:34:24 |

Last modified on | 2013-03-22 18:34:24 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 9 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 13A15 |

Classification | msc 13F05 |

Related topic | DedekindDomain |

Related topic | FractionalIdeal |

Related topic | PrimeIdeal |

Related topic | FundamentalTheoremOfIdealTheory |