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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A classification and examples of rank one chain domains


Authors: H. H. Brungs and N. I. Dubrovin
Journal: Trans. Amer. Math. Soc. 355 (2003), 2733-2753
MSC (2000): Primary 16L30, 16K40, 16W60; Secondary 20F29, 20F60
Published electronically: March 19, 2003
MathSciNet review: 1975397
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Abstract: A chain order of a skew field $D$ is a subring $R$ of $D$ so that $d\in D\backslash R$ implies $d^{-1}\in R.$ Such a ring $R$ has rank one if $J(R)$, the Jacobson radical of $R,$ is its only nonzero completely prime ideal. We show that a rank one chain order of $D$ is either invariant, in which case $R$ corresponds to a real-valued valuation of $D,$ or $R$ is nearly simple, in which case $R,$ $J(R)$ and $(0)$ are the only ideals of $R,$ or $R$ is exceptional in which case $R$ contains a prime ideal $Q$that is not completely prime. We use the group $\mathcal{M}(R)$ of divisorial $R{\text{-}ideals}$ of $D$ with the subgroup $\mathcal{H}(R)$ of principal $R{\text{-}ideals}$ to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index $k$of $\mathcal{H}(R) $ in $\mathcal{M}(R).$Using the covering group $\mathbb{G} $ of $\operatorname{SL}(2,\mathbb{R} )$ and the result that the group ring $T\mathbb{G} $ is embeddable into a skew field for $T$ a skew field, examples of rank one chain orders are constructed for each possible exceptional case.


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Additional Information

H. H. Brungs
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
Email: hbrungs@math.ualberta.ca

N. I. Dubrovin
Affiliation: Department of Mathematics, Vladimir State University, Gorki Str. 87, 600026 Vladimir, Russia
Email: ndubrovin@mail.ru

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03272-0
PII: S 0002-9947(03)03272-0
Keywords: Exceptional chain domains, skew field, valuation, cone, covering group.
Received by editor(s): April 10, 2002
Received by editor(s) in revised form: October 9, 2002
Published electronically: March 19, 2003
Additional Notes: The first author was supported by NSERC
The second author was supported by RFBR and DFG (grant no. 98-01-04110).
Article copyright: © Copyright 2003 American Mathematical Society