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 annihilator11In 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 –module 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 .
|Date of creation||2013-03-22 12:42:38|
|Last modified on||2013-03-22 12:42:38|
|Last modified by||djao (24)|