This paper presents a solution to a problem in the subject of rings of analytic functions. It was shown that [Bers (1948), Kakutani (1955)] two domains Dj and D2 in the complex plane were conformally equivalent (to within a certain equivalence relation) iff the rings B(DJ and B(D2) of ali bounded analytic functions defined on them were algebraically isomorphic. In the case of rectangles, two are conformally equivalent iff the ration of the sides of one equals the same ratio for the other [Uluçay (1946)]. It follows that this ratio must be contained somewhere in the algebraic structure of the ring. The problem is to find it.