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 \(R(D) \ne K\) $R(D)\ne K$ , then R(D) is a rank 1 discrete valuation ring that contains a field consisting of the units of D along with 0. Along the way, it is shown that R(D) contains all fractions of the form a/b, where \(a,b \in D \setminus \{0\}\) $a,b\in D\backslash \{0\}$ and a has Euclidean value no larger than that of b. Connections are also made to ideas from medieval Italian mathematics.