Ideal-theoretic characterizations of valuation and Prüfer monoids

Franz Halter-Koch

Archivum Mathematicum (2004)

  • Volume: 040, Issue: 1, page 41-46
  • ISSN: 0044-8753

Abstract

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It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen r -system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.

How to cite

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Halter-Koch, Franz. "Ideal-theoretic characterizations of valuation and Prüfer monoids." Archivum Mathematicum 040.1 (2004): 41-46. <http://eudml.org/doc/249286>.

@article{Halter2004,
abstract = {It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen $r$-system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.},
author = {Halter-Koch, Franz},
journal = {Archivum Mathematicum},
keywords = {valuation monoids; Prüfer domains; valuation monoids; Lorenzen -systems of ideals; Prüfer domains; root-closed cancellative monoids; Prüfer multiplication monoids; integral domains},
language = {eng},
number = {1},
pages = {41-46},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Ideal-theoretic characterizations of valuation and Prüfer monoids},
url = {http://eudml.org/doc/249286},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Halter-Koch, Franz
TI - Ideal-theoretic characterizations of valuation and Prüfer monoids
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 1
SP - 41
EP - 46
AB - It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen $r$-system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.
LA - eng
KW - valuation monoids; Prüfer domains; valuation monoids; Lorenzen -systems of ideals; Prüfer domains; root-closed cancellative monoids; Prüfer multiplication monoids; integral domains
UR - http://eudml.org/doc/249286
ER -

References

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  1. Aubert K. E., Some characterizations of valuation rings, Duke Math. J. 21 (1954), 517–525. (1954) MR0062727
  2. Garcia J. M., Jaros P., Santos E., Prüfer * -multiplication domains and torsion theories, Comm. Algebra 27 (1999), 1275–1295. (1999) MR1669156
  3. Halter-Koch F., Ideal Systems, Marcel Dekker 1998. (1998) Zbl0953.13001MR1828371
  4. Halter-Koch F.,, Construction of ideal systems having nice noetherian properties, Commutative Rings in a Non-Noetherian Setting (S. T. Chapman and S. Glaz, eds.), Kluwer 2000, 271–285. MR1858166
  5. Halter-Koch F., Characterization of Prüfer multiplication monoids and domains by means of spectral module systems, Monatsh. Math. 139 (2003), 19–31. Zbl1058.20049MR1981115
  6. Halter-Koch F., Valuation Monoids, Defining Systems and Approximation Theorems, Semigroup Forum 55 (1997), 33–56. (1997) Zbl0880.20047MR1446657

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