The unit fractions from a Euclidean domain generate a DVR

Abstract 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 $$R(D) \ne K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> , 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 $$a,b \in D \setminus \{0\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo>\</mml:mo> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and a has Euclidean value no larger than that of b . Connections are also made to ideas from medieval Italian mathematics.