
Table of contents for this issue  Previous article  Next article Leloup, Gérard
Cyclically valued rings and formal power series. Annales mathématiques Blaise Pascal, 14 no. 1 (2007), p. 3760
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Class. Math.: 13F25, 13A18, 13A99, 06F15, 06F99
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Rings of formal power series $k[[C]]$ with exponents in a cyclically ordered group $C$ were defined in [2]. Now, there exists a “valuation” on $k[[C]]$ : for every $\sigma $ in $k[[C]]$ and $c$ in $C$, we let $v(c,\sigma )$ be the first element of the support of $\sigma $ which is greater than or equal to $c$. Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in $k[[C]]$. We prove that a cyclically valued ring is a subring of a power series ring $k[[C,\theta ]]$ with twisted multiplication if and only if there exist invertible monomials of every degree, and the support of every element is wellordered. We also give a criterion for being isomorphic to a power series ring with twisted multiplication. Next, by the way of quotients of cyclic valuations, it follows that any power series ring $k[[C,\theta ]]$ with twisted multiplication is isomorphic to a $R^{\prime }[[C^{\prime },\theta ^{\prime }]]$, where $C^{\prime }$ is a subgroup of the cyclically ordered group of all roots of $1$ in the field of complex numbers, and $R^{\prime } \simeq k[[H,\theta ]]$, with $H$ a totally ordered group. We define a valuation $v(\epsilon ,\cdot )$ which is closer to the usual valuations because, with the topology defined by $v(a,\cdot )$, a cyclically valued ring is a topological ring if and only if $a=\epsilon $ and the cyclically ordered group is indeed a totally ordered one.
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Article  MR 6161  Zbl 0063.03135 [4] F.V. Kuhlmann, Valuation theory of fields, Preprint [5] G. Leloup, Existentially equivalent cyclically ultrametric distances and cyclic valuations, submitted2005 [6] S. Mac Lane, The uniqueness of the power series representation of certain fields with valuations, Annals of Mathematics, 39:370382, 1938 MR 1503414  Zbl 0019.04901  JFM 64.0970.05 [7] S. Mac Lane, The universality of formal power series fields, Bulletin of the American Mathematical Society, 45:888890, 1939 MR 610  Zbl 0022.30401  JFM 65.0093.02 [8] B. H. Neuman, On ordered division rings, Trans. Amer. Math. Soc., 66:202252, 1949 MR 32593  Zbl 0035.30401 [9] R. H. Redfield, Constructing latticeordered fields and division rings, Bull. Austral. Math. Soc., 40:365369, 1989 MR 1037630  Zbl 0683.12015 [10] P. Ribenboim, Théorie des Valuations, Les Presses de l’Université de Montréal, 1964 MR 249425  Zbl 0139.26201 
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