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Non-Commutative Valuation Rings and Semi-Hereditary Orders
- Publisher
- de Gruyter
- Copyright
- © de Gruyter 2009
- ISSN
- 0933-7741
- eISSN
- 1435-5337
- DOI
- 10.1515/FORUM.2009.007
- Publisher site
- See Article on Publisher Site
Abstract
We give an example of a commutative Prüfer domain R with field of fractions F and a quaternion division algebra D with centre F such that R cannot be extended to a Prüfer order in D in the sense of Alajbegović and Dubrovin, J. Algebra 135: 165–176, 1990. This shows, that a general extension theorem for Prüfer orders in central simple algebras does not exist and finally answers a question given in Marubayashi, Miyamoto, Ueda, Non-commutative Valuation Rings and Semihereditary Orders. K-Monographs in Mathematics 3, Kluwer, 1997. Moreover, in our example R is a Bézout domain which is the intersection of a countable number of (non-discrete) real valuation rings.
Journal
Forum Mathematicum – de Gruyter
Published: Jan 1, 2009