fractional ideal

1 Basics

Let A be an integral domainMathworldPlanetmath with field of fractionsMathworldPlanetmath K. Then K is an A–module, and we define a fractional idealMathworldPlanetmathPlanetmath of A to be a submodule of K which is finitely generatedMathworldPlanetmathPlanetmath as an A–module.

The product of two fractional ideals π”ž and π”Ÿ of A is defined to be the submodule of K generated by all the products xβ‹…y∈K, for xβˆˆπ”ž and yβˆˆπ”Ÿ. This product is denoted π”žβ‹…π”Ÿ, and it is always a fractional ideal of A as well. Note that, if A itself is considered as a fractional ideal of A, then π”žβ‹…A=π”ž. Accordingly, the set of fractional ideals is always a monoid under this product operation, with identity elementMathworldPlanetmath A.

We say that a fractional ideal π”ž is invertible if there exists a fractional ideal π”žβ€² such that π”žβ‹…π”žβ€²=A. It can be shown that if π”ž is invertible, then its inverseMathworldPlanetmathPlanetmathPlanetmath must be π”žβ€²=(A:π”ž), the annihilatorPlanetmathPlanetmath11In general, for any fractional ideals π”ž and π”Ÿ, the annihilator of π”Ÿ in π”ž is the fractional ideal (π”ž:π”Ÿ) consisting of all x∈K such that xβ‹…π”ŸβŠ‚π”ž. of π”ž in A.

2 Fractional ideals in Dedekind domains

We now suppose that A is a Dedekind domainMathworldPlanetmath. In this case, every nonzero fractional ideal is invertible, and consequently the nonzero fractional ideals in A form a group under ideal multiplication, called the ideal group of A.

The unique factorizationMathworldPlanetmath of ideals theorem states that every fractional ideal in A factors uniquely into a finite product of prime idealsMathworldPlanetmathPlanetmathPlanetmath of A and their (fractional ideal) inverses. It follows that the ideal group of A is freely generated as an abelian groupMathworldPlanetmath by the nonzero prime ideals of A.

A fractional ideal of A is said to be principal if it is generated as an A–module by a single element. The set of nonzero principal fractional ideals is a subgroupMathworldPlanetmathPlanetmath of the ideal group of A, and the quotient groupMathworldPlanetmath of the ideal group of A by the subgroup of principal fractional ideals is nothing other than the ideal class groupPlanetmathPlanetmathPlanetmath of A.

Title fractional ideal
Canonical name FractionalIdeal
Date of creation 2013-03-22 12:42:38
Last modified on 2013-03-22 12:42:38
Owner djao (24)
Last modified by djao (24)
Numerical id 5
Author djao (24)
Entry type Definition
Classification msc 13A15
Classification msc 13F05
Related topic IdealClassGroup
Defines ideal group