Residue class rings of real-analytic and entire functions

Marek Golasiński ¹   ; Melvin Henriksen ²

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland
² Department of Mathematics Harvey Mudd College Clarement, CA 91711, U.S.A.

Colloquium Mathematicum, Tome 104 (2006) no. 1, pp. 85-97

Résumé

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.

Détail

DOI : 10.4064/cm104-1-5

Keywords: mathcal mathbb mathcal mathbb denote respectively ring analytic real entire functions variable shown mathfrak maximal ideal mathcal mathbb mathcal mathbb mathfrak isomorphic either reals real closed field eta set while mathfrak maximal ideal mathcal mathbb mathcal mathbb mathfrak isomorphic latter fields field complex numbers moreover study residue class rings prime ideals these rings their krull dimensions made classical characterization algebraically closed fields due nbsp steinitz techniques described nbsp gillman nbsp jerisons book rings continuous functions