If F is an open Riemann surface
and A(F) is the set of all analytic functions on F, then A(F) is a ring under
pointwise addition and multiplication. This paper is concerned with proper subrings
R of A(F) which are isomorphic images of A(G), the ring of all analytic
functions on an open Riemann surface G, under a homomorphism Φ which
maps constant functions onto themselves. The ring R has the form :
g ∈ A(G), ϕ an analytic map from F into , and will be denoted Rϕ. Relations
between ϕ, Rϕ and the spectrum of Rcβ are given as necessary and sufficient
conditions for the existence of a Riemann surface G such that R is isomorphic to
A(G).
:
, and will be denoted 