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Search: All articles in the CMB digital archive with keyword valuations

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1. CMB 2010 (vol 54 pp. 381)

Velušček, Dejan
A Short Note on the Higher Level Version of the Krull--Baer Theorem
Klep and Velu\v{s}\v{c}ek generalized the Krull--Baer theorem for higher level preorderings to the non-commutative setting. A $n$-real valuation $v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section of $\overline{v}$ is a crucial ingredient of the construction of a complete preordering on the base field $D$ such that its projection on the residue skew field $k_v$ equals the given level $1$ ordering on $k_v$. In the article we give a proof of the existence of the section of $\overline{v}$, which was left as an open problem by Klep and Velu\v{s}\v{c}ek, and thus complete the generalization of the Krull--Baer theorem for preorderings.

Keywords:orderings of higher level, division rings, valuations
Categories:14P99, 06Fxx

2. CMB 2007 (vol 50 pp. 105)

Klep, Igor
On Valuations, Places and Graded Rings Associated to $*$-Orderings
We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

Keywords:$*$--orderings, valuations, rings with involution
Categories:14P10, 16S30, 16W10

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