The unit fractions from a Euclidean domain generate a DVR

Let D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if R(D) is not equal to K, then R(D) is a rank 1 discrete valuation ring that contains a field consisting of the units of D along with 0. Connections are made to ideas from medieval Italian mathematics.