fractional ideal

Ordinary ideals of a ring are sets of multiples that are closed under addition; they live inside the ring. To build a group under multiplication, though, we need inverses, and ideals as usually defined have none. The fix is to allow a single common denominator, just as 3/4 lets you build fractions out of integers. A fractional ideal is an “ideal with a bounded denominator,” and these do form a group.

Let R be a Dedekind domain (for instance O_K) with fraction field K. A fractional ideal is a nonzero finitely generated R-submodule M of K. Equivalently, there is a nonzero d in R with dM contained in R, so M = (1/d) I for an ordinary ideal I. The product of two fractional ideals is generated by all products of their elements, and the ordinary ideals are exactly the fractional ideals contained in R, the integral ones.

The decisive theorem is that in a Dedekind domain every nonzero fractional ideal is invertible: M times M^{-1} = R, where M^{-1} = {x in K : xM is contained in R}. Hence the nonzero fractional ideals form an abelian group under multiplication, with identity R. The principal fractional ideals (aR for a in K^*) form a subgroup, and the quotient is the ideal class group.