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fractional ideal (Definition)

Basics

Let $ A$ be an integral domain with field of fractions $ K$. Then $ K$ is an $ A$-module, and we define a fractional ideal of $ A$ to be a submodule of $ K$ which is finitely generated as an $ A$-module.

The product of two fractional ideals $ {\mathfrak{a}}$ and $ \b$ of $ A$ is defined to be the submodule of $ K$ generated by all the products $ x \cdot y \in K$, for $ x \in {\mathfrak{a}}$ and $ y \in \b$. This product is denoted $ {\mathfrak{a}} \cdot \b$, 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 $ {\mathfrak{a}}\cdot A = {\mathfrak{a}}$. Accordingly, the set of fractional ideals is always a monoid under this product operation, with identity element $ A$.

We say that a fractional ideal $ {\mathfrak{a}}$ is invertible if there exists a fractional ideal $ {\mathfrak{a}}'$ such that $ {\mathfrak{a}}\cdot {\mathfrak{a}}' = A$. It can be shown that if $ {\mathfrak{a}}$ is invertible, then its inverse must be $ {\mathfrak{a}}' = (A:{\mathfrak{a}})$, the annihilator 1 of $ {\mathfrak{a}}$ in $ A$.

Fractional ideals in Dedekind domains

We now suppose that $ A$ is a Dedekind domain. 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 factorization of ideals theorem states that every fractional ideal in $ A$ factors uniquely into a finite product of prime ideals of $ A$ and their (fractional ideal) inverses. It follows that the ideal group of $ A$ is freely generated as an abelian group 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 subgroup of the ideal group of $ A$, and the quotient group of the ideal group of $ A$ by the subgroup of principal fractional ideals is nothing other than the ideal class group of $ A$.


Footnotes

...1
In general, for any fractional ideals $ {\mathfrak{a}}$ and $ \b $, the annihilator of $ \b $ in $ {\mathfrak{a}}$ is the fractional ideal $ ({\mathfrak{a}}:\b )$ consisting of all $ x \in K$ such that $ x\cdot\b \subset {\mathfrak{a}}$.


"fractional ideal" is owned by djao.
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See Also: ideal class group

Also defines:  ideal group
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Cross-references: ideal class group, quotient group, subgroup, abelian group, freely generated, prime ideals, finite, factors, multiplication, ideal, group, Dedekind domain, annihilator, inverse, invertible, identity element, operation, monoid, generated by, product, finitely generated, submodule, field of fractions, integral domain
There are 14 references to this entry.

This is version 2 of fractional ideal, born on 2002-06-02, modified 2002-06-02.
Object id is 2995, canonical name is FractionalIdeal.
Accessed 6356 times total.

Classification:
AMS MSC: 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
 13F05 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Dedekind, Prüfer and Krull rings and their generalizations)
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