Residue class rings of real-analytic and entire functions

Marek Golasiński, and Melvin Henriksen. "Residue class rings of real-analytic and entire functions." Colloquium Mathematicae 104.1 (2006): 85-97. <http://eudml.org/doc/286080>.

@article{MarekGolasiński2006,
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 η₁-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.},
author = {Marek Golasiński, Melvin Henriksen},
journal = {Colloquium Mathematicae},
keywords = {Bézout domain; Krull dimension; real-analytic (entire) function; real closed field; integral closure; maximal ideals},
language = {eng},
number = {1},
pages = {85-97},
title = {Residue class rings of real-analytic and entire functions},
url = {http://eudml.org/doc/286080},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Marek Golasiński
AU - Melvin Henriksen
TI - Residue class rings of real-analytic and entire functions
JO - Colloquium Mathematicae
PY - 2006
VL - 104
IS - 1
SP - 85
EP - 97
AB - 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 η₁-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.
LA - eng
KW - Bézout domain; Krull dimension; real-analytic (entire) function; real closed field; integral closure; maximal ideals
UR - http://eudml.org/doc/286080
ER -