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.