A fractional ideal is a generalization of an ideal in a ring R. Instead, a fractional ideal is contained in the number field F, but has the property that there is an element such that

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

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

For example, in the field , the set

	(3)

is a fractional ideal because

Note that , where

	(4)

and so is an inverse to .

Given any fractional ideal there is always a fractional ideal such that . Consequently, the fractional ideals form an Abelian group by multiplication. The principal ideals generate a subgroup P, and the quotient group is called the ideal class group.

 

Class Group, Grothendieck Group, Ideal, Number Field