  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.
