First we verify that defines an equivalence relation on , where the only condition that isn't immediately obvious is transitivity. Suppose so that

Then , so , as required.

Next we show that defines a congruence relation on the monoid . Now suppose such that and . Then so . The quotient monoid is thence well-defined.

Now we show that is in fact a group. Let and . Then so for some ideal .

Finally we show that these groups are isomorphic, letting and denote the constructions with and without fraction ideals respectively. An arbitrary element in is for some fractional ideal . But for some and , so . Therefore we can always use an integral ideal as a representative for an element of , which itself represents an element of . Clearly : hence we have an isomorphism.