Global ideal theory of meromorphic function fields

Abstract

Abstract. 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.

Rings which resemble rings of entire functions

“…The following lemma is a straightforward extension of a well-known result on rings of functions [4,10]. [4] an ideal / of a ring R is called local if it is contained in a unique maximal ideal.…”

“…The rest is given in [4, §2] for rings of analytic functions and easily extends to arbitrary Bezout domains. As an example we consider part (a) which appears in [4] to be the least straightforward. First note that if J, J' are dual ideals of a lattice ordered abelian group G, then JAJ' is a dual ideal [4, Lemma 2.8].…”

“…for more general W-rings R, and also information about the valuation theory of such rings, by using the correspondence between ideals of R and the A-filters of X(R). We adapt the notation of [4] to our setting. If R is a W-ring and re R, let Z(r) = {Pe X(R) \reP}.…”

“…[4] an ideal / of a ring R is called local if it is contained in a unique maximal ideal. In [2,4] the decompositions of ideals of A(X) into local ideals were studied and in [3,20] the primary ideals of A(X) were studied where X is a non-compact connected Riemann surface. We add a few remarks on these ideas.…”

“…In §2 we consider the group of divisibility and indicate how results of Ailing [1], [2], [3], [4] on the ideal theory and valuation theory of meromorphic function fields can be extended to Klingen's more abstract setting. We conclude in §3 with some remarks on realizing a W-ring as a ring of analytic functions on a Riemann surface.…”