A fractional ideal is a generalization of an ideal in a ring . Instead, a fractional ideal is contained in the number field , but has the property that there is an 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, in the field , the set
(3)
is a fractional ideal because
(4)
Note that , where
(5)
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 , and the quotient group is called the ideal class group.
. Instead, a fractional ideal is contained in the number field 
, but has the property that there is an element 
such that
. In particular, every element in 
can be written as a fraction, with a fixed denominator.
, the set
, where
is an inverse to 
.
there is always a fractional ideal 
such that 
. Consequently, the fractional ideals form an Abelian group by multiplication. The principal ideals generate a subgroup 
, and the quotient group is called the ideal class group.