Ideal theory on open Riemann surfaces

Abstract

Introduction. The theorems of the classical ideal theory in fields of algebraic numbers hold in rings of analytic functions on compact Riemann surfaces. The surfaces admitted in our discussion are closely related to algebraic surfaces; we deal either with compact surfaces from which a finite number of points are omitted or, more generally, with surfaces determined by an algebroid function. The local aspects of the resulting ideal theory are the same as those found in the theory of algebraic functions. However, the ideal theory in the large is quite different. We now cannot count on the simplifications which are implied by the ascending chain condition for ideals. The recent theories of functions on topological spaces provide the necessary tools for a simple theory in the large. We shall show that the topologization of rings of entire functions by means of the topology of the underlying surface furnishes a fruitful method. Thus, the closed maximal ideals will correspond to the points of the given surfaces. Finally, to quote another result, all closed ideals are principal.

Residue class rings of real-analytic and entire functions

“…In [16] Schilling claimed to have shown that K-dim E(C) = 1, but in 1952 Kaplansky showed that it is at least 2; then Henriksen proved [9] that it is at least 2 ω 1 and also discussed the nature of the residue class rings E(C)/p, where p is a prime ideal of E(C).…”

Global ideal theory of meromorphic function fields

“…The author was surprised to learn (1.3) that X0 and Y0 are always homeomorphic. One possible inference to be drawn is Theorem [18], that states that every closed ideal of A is principal, is generalized from C to X (5.14). The final topic to be discussed in §5 is that of subrings B of 5 that contain A, which Kelleher [15] called A-rings.…”

Global Ideal Theory of Meromorphic Function Fields

“…Let v map / G F(X) to its divisor in (Div X) u {oo}. In 1946 Schilling [18] considered fractional ideals in F(C). Generalizing his work we consider, in §2, the set of all sub .¿-modules of 5, sam (5); the set of all fractional ideals of 5,1(F); and the set of all ideals of A, 1(A).…”

“…Theorem [18], that states that every closed ideal of A is principal, is generalized from C to X (5.14). The final topic to be discussed in §5 is that of subrings B of 5 that contain A, which Kelleher [15] called A-rings.…”

“…Initially the concern of the researchers in this field was with the ideal theory of A(C). In 1946 Schilling [18] considered fractional ideals of 5(C). We will generalize this in two directions by considering {sub-^4-modules of F(X)} = sam(5(Ar)) (or sam(5) for short).…”