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.
Let D be a division ring of fractions of a crossed product , 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 of all formal power series in G over
F with respect to . From this we obtain that all division rings of fractions of
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, possesses a division
ring of fraction which is free in this sense if and only if the rational closure of in
the endomorphism ring of the corresponding right F-vector space is a skew field.
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.