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.
@article{bwmeta1.element.bwnjournal-article-aav64i2p125bwm,
author = {Clifford S. Queen},
title = {A simple characterization of principal ideal domains},
journal = {Acta Arithmetica},
volume = {64},
year = {1993},
pages = {125-128},
zbl = {0780.13012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav64i2p125bwm}
}
Clifford S. Queen. A simple characterization of principal ideal domains. Acta Arithmetica, Tome 64 (1993) pp. 125-128. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav64i2p125bwm/
[000] [1] H. Cohn, Advanced Number Theory, Dover, 1980. | Zbl 0474.12002
[001] [2] N. Jacobson, Basic Algebra I, 2nd ed., Freeman, 1985. | Zbl 0557.16001
[002] [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.
[003] [4] T. Motzkin, The Euclidean algorithm, Bull. Amer. Math. Soc. 55 (1949), 1142-1146. | Zbl 0035.30302
[004] [5] C. Queen, Arithmetic euclidean rings, Acta Arith. 26 (1974), 105-113. | Zbl 0423.12013
[005] [6] G. Rabinowitsch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern, J. Reine Angew. Math. 142 (1913), 153-164. | Zbl 44.0243.03
[006] [7] P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301. | Zbl 0223.13019