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  • Author: Joachim Gräter x
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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.

Abstract

Let D be a division ring of fractions of a crossed product F[G,η,α], where F is a skew field and G is a group with Conradian left-order . For D we introduce the notion of freeness with respect to and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F((G)) of all formal power series in G over F with respect to . From this we obtain that all division rings of fractions of F[G,η,α] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F[G,η,α] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G,η,α] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field.

Abstract

We present and discuss methods for embedding a left-ordered group G into the multiplicative group of a skew field D. Among others, we are interested in the case in which any group automorphism of G extends to a ring automorphism of D and we will be able to apply Dubrovin's embedding theorem to a special class of left-ordered groups which we call locally complete.