Some curiosities of rings of analytic functions

Abstract

1. It is the purpose of this paper to exhibit some properties of certain rings of analytic functions which may be a little unexpected. Let E be the ring of all entire functions in one complex variable, i.e. the subring of C[[X]] consisting of all formal power series with infinite convergence radius. More generally, for a subfield K of C let E(K) be the subring of K[[X]] formed by all power series with infinite convergence radius. If Q is a positive real number, let E(Q, K) , resp. J?(Q, K) be the subring of K[[X]] consisting of all power series with convergence radius >Q, resp. 2~. It is well known that E(K) is a non-Noetherian domain and that E(K) is a Bezout domain, i.e. any finitely generated ideal is principal. If K=C this was proved by Wedderburn [ 131 and in the general case by Helmer [4], (who apparently was unaware of [13]). As for the Krull dimension of E the first ‘result’ appeared in [lo] stating that K-dim E = 1. An error in the proof was noticed by Kaplansky [5] and K-dim E is actually infinite. We shall give more precise results concerning the length of chains of prime ideals of E. As shown in [7] the global dimension of E is 23, while the exact value of gl.dimE cannot be determined from the usual axioms of set theory (ZFC): For any t, 3 5 tr m, the statement gl.dim E = t is consistent with ZFC, in fact, even consistent with ZFC + MA, (MA denoting Martin’s axiom). The corresponding results hold true if E is replaced by E(K) or &, K). The proofs only require minor modifications. For E(Q, K), however, the situation is completely different. For any positive Q and any field KS C the ring E(Q, K) is Euclidean, in particular a PID. Since for instance E(1, C) = n;=, E(l l/n, 4X) we obtain a decreasing sequence of PID’s whose intersection is a Bezout domain of undecidable global dimension and of uncountable Krull dimension. The stable range (in the sense of Bass [l]) of the above rings depends on K. If KS R the stable range of each of the rings E(Q, K), E(Q, K) and E(K) is 2, otherwise, when Kg R the stable range is 1.