Curves and coherent Prüfer rings

We show, in constructive mathematics, that if k is a discrete field and f an arbitrary polynomial in k[x,y] then the localisation Rfy is always a semihereditary ring, where R denotes the ring k[x,y] quotiented by f. An important corollary is that R is semiherditary whenever 1=〈f,fx,fy〉. This can be seen as the constructive content of the theorem saying that if moreover R is a domain, then it is Dedekind.