ON RINGS WITH A CERTAIN TYPE OF FACTORIZATION AND COMPACT RIEMANN SURFACES

Let V be a compact Riemann surface, V' be the complement of a nonvoid finite subset of V and A(V') be the ring of finite sums of meromorphic functions in V' with finite divisor. In this paper it is proved that every nonzero f element-of A(V') can be decomposed as a product alpha-beta, where-alpha is either a unit or a product of powers of irreducible elements of A(V'), uniquely determined by f up to multiplication by units, and beta is a product of functions of the type e-phi - 1, with phi-holomorphic and nonconstant in V'. Furthermore, a similar result is obtained for a certain class of subrings of A(V').