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 a_n/a_d (numerator over denominator), and b as b_n/b_d. 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) =
a_db_d aw(b) over a_db_d =
b_da_nw(b) over a_db_d =
b_dw(a_nb) over a_db_d =
w(a_nb_n) over a_db_d

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.