# A simple characterization of principal ideal domains

Acta Arithmetica (1993)

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

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topClifford 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

top- [1] H. Cohn, Advanced Number Theory, Dover, 1980. Zbl0474.12002
- [2] N. Jacobson, Basic Algebra I, 2nd ed., Freeman, 1985. Zbl0557.16001
- [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] T. Motzkin, The Euclidean algorithm, Bull. Amer. Math. Soc. 55 (1949), 1142-1146. Zbl0035.30302
- [5] C. Queen, Arithmetic euclidean rings, Acta Arith. 26 (1974), 105-113. Zbl0423.12013
- [6] G. Rabinowitsch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern, J. Reine Angew. Math. 142 (1913), 153-164. Zbl44.0243.03
- [7] P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301. Zbl0223.13019

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