Residue class rings of real-analytic and entire functions

Abstract

Abstract. Let A(R) and E(R) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real closed field that is an η 1 -set, while if m is a maximal ideal of E(R), then E(R)/m is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of a classical characterization of algebraically closed fields due to E. Steinitz and techniques described in L. Gillman and M. Jerison's book on rings of continuous functions.

The Ziegler Spectrum of the Ring of Entire Complex Valued Functions

“…In particular fixed prime ideals are exactly the maximal ideals M t = (z − t)E, t ∈ C. If P is not fixed, then, because all calculations are made modulo a nonprincipal ultrafilter U , the property of being prime is quite tricky. For instance (see [5]) for each pair of prime ideals P ⊂ P there exist at least 2 ℵ 1 ideals strictly between P and P . The main idea is that this interval contains a Dedekind complete 1 -set of prime ideals, hence [4,Corollary 13.24] gives the desired cardinality.…”

“…Fact 2.2. (see [6,Prop. 1.1]) Let {z k } be an absolute value nondecreasing sequence of complex numbers with no finite accumulation point.…”