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Let R be the ring of entire functions, and let K be the complex field. The ring R consists of all functions from K to K differentiable everywhere (in the usual sense).

The algebraic structure of the ring of entire functions seems to have been investigated extensively first by O. Helmer [1].

The ideals of R are herein classified as in [2]: an ideal I is called fixed if every function in it vanishes at at least one common point; otherwise, I is called free. The structure of the fixed ideals was determined in [1]. The structure of the free ideals is determined below.

While examples of free ideals are easily given, transfinite methods seem to be needed to construct maximal free ideals. The latter are characterized below, and it is shown that the residue class field of a maximal free ideal is always isomorphic to K; the field theory of E. Steinitz [5] is used.


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