Residue class rings of real-analytic and entire functions

Tom 104 / 2006

Marek Golasiński, Melvin Henriksen Colloquium Mathematicum 104 (2006), 85-97 MSC: Primary 26E05, 30D20; Secondary 12D15, 13B30. DOI: 10.4064/cm104-1-5


Let $\mathcal{A}(\mathbb{R})$ and $\mathcal{E}(\mathbb{R})$ denote respectively the ring of analytic and real entire functions in one variable. It is shown that if $\mathfrak{m}$ is a maximal ideal of $\mathcal{A}(\mathbb{R})$, then $\mathcal{A}(\mathbb{R})/\mathfrak{m}$ is isomorphic either to the reals or a real closed field that is an $\eta_1$-set, while if $\mathfrak{m}$ is a maximal ideal of $\mathcal{E}(\mathbb{R})$, then $\mathcal{E}(\mathbb{R})/\mathfrak{m}$ 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.


  • Marek GolasińskiFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
  • Melvin HenriksenDepartment of Mathematics
    Harvey Mudd College
    Clarement, CA 91711, U.S.A.

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