Affiliation:
1. Institut für Mathematik , Universität Potsdam , Karl-Liebknecht-Straße 24–25, 14476 Potsdam OT Golm , Germany
Abstract
Abstract
Let D be a division ring of fractions of a crossed product
F
[
G
,
η
,
α
]
{F[G,\eta,\alpha]}
, where F is a skew field
and G is a group with Conradian left-order
≤
{\leq}
. For D we introduce the notion of freeness with
respect to
≤
{\leq}
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
)
)
{F((G))}
of all formal power series in G over
F with respect to
≤
{\leq}
. From this we obtain that all division rings of fractions of
F
[
G
,
η
,
α
]
{F[G,\eta,\alpha]}
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
]
{F[G,\eta,\alpha]}
possesses a division
ring of fraction which is free in this sense if and only if the rational closure of
F
[
G
,
η
,
α
]
{F[G,\eta,\alpha]}
in
the endomorphism ring of the corresponding right F-vector space
F
(
(
G
)
)
{F((G))}
is a skew field.
Subject
Applied Mathematics,General Mathematics
Reference30 articles.
1. A. A. Bovdi,
Crossed products of a semigroup and a ring,
Dokl. Akad. Nauk SSSR 137 (1961), 1267–1269.
2. S. D. Brodskiĭ,
Equations over groups, and groups with one defining relation,
Sibirsk. Mat. Zh. 25 (1984), no. 2, 84–103.
3. R. G. Burns and V. W. D. Hale,
A note on group rings of certain torsion-free groups,
Canad. Math. Bull. 15 (1972), 441–445.
4. A. Clay and D. Rolfsen,
Ordered Groups and Topology,
Grad. Stud. Math. 176,
American Mathematical Society, Providence, 2016.
5. P. M. Cohn,
An Introduction to Ring Theory,
Springer Undergrad. Math. Ser.,
Springer, London, 2000.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献