Su (1972) proved that for auy two suhsets X, Y of C, the complex plane, X and Y are conformally homeomorphic if there is an isomorphîsm between H(X) and H(Y) which is the identity on constant functions. Minda (1976) extended the method to the rings of holomorphic functions on any suhsets of öpen Riemann surfaces. Royden (1963) listed some properties which a ring of functions might have, In this paper, using the results of Su, Minda, and Royden we present some properties of the subring R