fractional ideal
1 Basics
Let be an integral domain with field of fractions
. Then is
an βmodule, and we define a fractional ideal
of to be a
submodule of which is finitely generated
as an βmodule.
The product of two fractional ideals and of is defined
to be the submodule of generated by all the products , for and . This product is denoted , and it is always a fractional ideal of as well. Note
that, if itself is considered as a fractional ideal of , then
. Accordingly, the set of fractional ideals is always
a monoid under this product operation, with identity element .
We say that a fractional ideal is invertible if there
exists a fractional ideal such that . It can
be shown that if is invertible, then its inverse must be , the annihilator
11In general, for any fractional
ideals and , the annihilator of in is the
fractional ideal consisting of all such that
. of in .
2 Fractional ideals in Dedekind domains
We now suppose that is a Dedekind domain. In this case, every
nonzero fractional ideal is invertible, and consequently the nonzero
fractional ideals in form a group under ideal multiplication,
called the ideal group of .
The unique factorization of ideals theorem states that every
fractional ideal in factors uniquely into a finite product of
prime ideals
of and their (fractional ideal) inverses. It follows
that the ideal group of is freely generated as an abelian group
by
the nonzero prime ideals of .
A fractional ideal of is said to be principal if it is
generated as an βmodule by a single element. The set of nonzero
principal fractional ideals is a subgroup of the ideal group of ,
and the quotient group
of the ideal group of by the subgroup of
principal fractional ideals is nothing other than the ideal class
group
of .
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 |