Global Ideal Theory of Meromorphic Function Fields

It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved. Introduction. Throughout this paper let X and Y denote noncompact (connected) Riemann surfaces. Let A(X) (or A for short), denote the ring of all analytic functions on X, and let F(X) (or 5 for short), denote the field of all meromorphic functions on X. In 1940 Helmer [10] studied divisibility properties in .4(C), hiid the foundations for its ideal theory, and proved that every finitely generated ideal in it is principal. (See [2, pp. 24-28] for a brief history of the subject from 1940 to 1966.) In 1952-53 Henriksen [11], [12] investigated the maximal and prime ideals of A(C), finding-among other things-that each prime ideal is contained in a unique maximal ideal. An ideal of a ring will be called local if it is contained in a unique maximal ideal; thus Henriksen proved that each prime ideal of .4(C) is local. In 1948 Florack [7] proved essentially that X is a Stein manifold. Using her theorem, the investigation of the ideal theory of A(X), for X c C, was gradually generalized to arbitrary X. In 1963 the author [1] showed that if M is a maximal ideal of A then the ring of quotients, AM, is a valuation ring. At that time the value group of AM was also investigated. Using classical methods of commutative algebra, one can make a very complete analysis of the local ideals of A. The initial aim of this research was to learn more about the decomposition of an ideal I of A as an intersection of local ideals. In trying to extend local knowledge to obtain global results it became evident that some topology on the set specm A of maximal ideals was needed. The author turned, naturally, to the Zariski topology on specm A. X is, in a natural way, identifiable with a subset of specm A. Let X0 be the topology induced on X by this identification; it will be called the zero set topology on X. It is obvious that X0 is a much coarser topology than X. The author was surprised to learn (1.3) that X0 and Y0 are always homeomorphic. One possible inference to be drawn is Received by the editors December 1, 1978. AMS (MOS) subject classifications (1970). Primary 13A15, 30A98.