Dedekind Domains, Fractions and R Homomorphisms

Fractions and R Homomorphisms

Let R be an integral domain with fraction field F.  Let w be a module homomorphism from a fractional ideal H into F.  We will show that a*w(b) = b*w(a).  This certainly holds if a and b come from R, whence aw(b) = w(ab) = bw(a).  That's the definition of an R module homomorphism.  But the relationship extends to all of F.

Write a as an/ad (numerator over denominator), and b as bn/bd.  Follow along as we do some algebra.  The second step multiplies top and bottom by something in R, and this works only because R is an integral domain.

aw(b) =
adbd aw(b) over adbd =
bdanw(b) over adbd =
bdw(anb) over adbd =
w(anbn) over adbd

The last expression is symmetric in a and b, so the same algebra applies to bw(a), and we have aw(b) = bw(a).

This assumes a and b are both in H.