Bezout domains and elliptic curves

Let k be a fixed algebraic closure of and k(t)ac a fixed algebraic closure of k(t). Let Sk[t]\ {0} be a multiplicative set. Let A=S-1(k[t]) and [image omitted] be the integral closure of A in k(t)ac. We use elliptic curves to develop a necessary condition on S for [image omitted] to be a Bezout domain. We give some examples of S which fail to satisfy this condition. As a consequence, we eliminate some candidates for a good Rumely domain of characteristic 0 with algebraic subring k.