# ideal decomposition in Dedekind domain

According to the entry “fractional ideal^{} (http://planetmath.org/FractionalIdeal)”, we can that in a Dedekind domain^{} $R$, each non-zero integral ideal $\U0001d51e$ may be written as a product of finitely many prime ideals^{} ${\U0001d52d}_{i}$ of $R$,

$$\U0001d51e={\U0001d52d}_{1}{\U0001d52d}_{2}\mathrm{\dots}{\U0001d52d}_{k}.$$ |

The product decomposition is unique up to the order of the factors. This is stated and proved, with more general assumptions, in the entry “prime ideal factorisation is unique (http://planetmath.org/PrimeIdealFactorizationIsUnique)”.

Corollary. If ${\alpha}_{1}$, ${\alpha}_{2}$, …, ${\alpha}_{m}$ are elements of a Dedekind domain $R$ and $n$ is a positive integer, then one has

${({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{m})}^{n}=({\alpha}_{1}^{n},{\alpha}_{2}^{n},\mathrm{\dots},{\alpha}_{m}^{n})$ | (1) |

for the ideals of $R$.

This corollary may be proven by induction on the number $m$ of the $n$).

Title | ideal decomposition in Dedekind domain |

Canonical name | IdealDecompositionInDedekindDomain |

Date of creation | 2015-05-05 19:05:43 |

Last modified on | 2015-05-05 19:05:43 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 11R37 |

Classification | msc 11R04 |

Related topic | ProductOfFinitelyGeneratedIdeals |

Related topic | PolynomialCongruence |

Related topic | CancellationIdeal |

Related topic | DivisibilityInRings |

Related topic | IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal |

Related topic | DivisorTheory |

Related topic | GreatestCommonDivisorOfSeveralIntegers |