Fractional Ideal

A fractional ideal of an integral domain is an -submodule of its field of fractions , but has the property that there is a nonzero element such that

(1)

is an ideal in . In particular, every element in can be written as a fraction, with a fixed denominator.

(2)

Note that the multiplication of two fractional ideals is another fractional ideal.

For example, let in the number field , and put

(3)

Its inverse fractional ideal is

(4)

Indeed, is an ideal of , and

(5)

because the product contains both 2 and 3 and therefore contains 1.

When is a Dedekind ring, every nonzero fractional ideal has a fractional ideal inverse satisfying . Consequently, the nonzero fractional ideals form an Abelian group under multiplication. The principal ideals generate a subgroup , and the quotient group is called the class group.