A simple characterization of principal ideal domains

Clifford S. Queen

Acta Arithmetica (1993)

  • Volume: 64, Issue: 2, page 125-128
  • ISSN: 0065-1036

Abstract

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1. Introduction. In this note we give necessary and sufficient conditions for an integral domain to be a principal ideal domain. Curiously, these conditions are similar to those that characterize Euclidean domains. In Section 2 we establish notation, discuss related results and prove our theorem. Finally, in Section 3 we give two nontrivial applications to real quadratic number fields.

How to cite

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Clifford S. Queen. "A simple characterization of principal ideal domains." Acta Arithmetica 64.2 (1993): 125-128. <http://eudml.org/doc/206541>.

@article{CliffordS1993,
abstract = {1. Introduction. In this note we give necessary and sufficient conditions for an integral domain to be a principal ideal domain. Curiously, these conditions are similar to those that characterize Euclidean domains. In Section 2 we establish notation, discuss related results and prove our theorem. Finally, in Section 3 we give two nontrivial applications to real quadratic number fields.},
author = {Clifford S. Queen},
journal = {Acta Arithmetica},
keywords = {principal ideal domains},
language = {eng},
number = {2},
pages = {125-128},
title = {A simple characterization of principal ideal domains},
url = {http://eudml.org/doc/206541},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Clifford S. Queen
TI - A simple characterization of principal ideal domains
JO - Acta Arithmetica
PY - 1993
VL - 64
IS - 2
SP - 125
EP - 128
AB - 1. Introduction. In this note we give necessary and sufficient conditions for an integral domain to be a principal ideal domain. Curiously, these conditions are similar to those that characterize Euclidean domains. In Section 2 we establish notation, discuss related results and prove our theorem. Finally, in Section 3 we give two nontrivial applications to real quadratic number fields.
LA - eng
KW - principal ideal domains
UR - http://eudml.org/doc/206541
ER -

References

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  1. [1] H. Cohn, Advanced Number Theory, Dover, 1980. Zbl0474.12002
  2. [2] N. Jacobson, Basic Algebra I, 2nd ed., Freeman, 1985. Zbl0557.16001
  3. [3] M. Kutsuna, On a criterion for the class number of a real quadratic field to be one, Nagoya Math. J. 79 (1980), 123-129. 
  4. [4] T. Motzkin, The Euclidean algorithm, Bull. Amer. Math. Soc. 55 (1949), 1142-1146. Zbl0035.30302
  5. [5] C. Queen, Arithmetic euclidean rings, Acta Arith. 26 (1974), 105-113. Zbl0423.12013
  6. [6] G. Rabinowitsch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern, J. Reine Angew. Math. 142 (1913), 153-164. Zbl44.0243.03
  7. [7] P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301. Zbl0223.13019

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