Cyclically valued rings and formal power series
Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1 , p. 37-60
doi : 10.5802/ambp.226
URL stable : http://www.numdam.org/item?id=AMBP_2007__14_1_37_0

Classification:  13F25,  13A18,  13A99,  06F15,  06F99
Rings of formal power series $k\left[\left[C\right]\right]$ with exponents in a cyclically ordered group $C$ were defined in [2]. Now, there exists a “valuation” on $k\left[\left[C\right]\right]$ : for every $\sigma$ in $k\left[\left[C\right]\right]$ and $c$ in $C$, we let $v\left(c,\sigma \right)$ 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 $k\left[\left[C\right]\right]$. We prove that a cyclically valued ring is a subring of a power series ring $k\left[\left[C,\theta \right]\right]$ with twisted multiplication if and only if there exist invertible monomials of every degree, and the support of every element is well-ordered. 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 $k\left[\left[C,\theta \right]\right]$ with twisted multiplication is isomorphic to a ${R}^{\prime }\left[\left[{C}^{\prime },{\theta }^{\prime }\right]\right]$, where ${C}^{\prime }$ is a subgroup of the cyclically ordered group of all roots of $1$ in the field of complex numbers, and ${R}^{\prime }\simeq k\left[\left[H,\theta \right]\right]$, with $H$ a totally ordered group. We define a valuation $v\left(ϵ,·\right)$ which is closer to the usual valuations because, with the topology defined by $v\left(a,·\right)$, a cyclically valued ring is a topological ring if and only if $a=ϵ$ and the cyclically ordered group is indeed a totally ordered one.

### Bibliographie

[1] Fuchs, L. Partially Ordered Algebraic Structures, Pergamon Press (1963) Zbl 0137.02001

[2] Giraudet, M.; Kuhlmann, F.-V.; Leloup, G. Formal power series with cyclically ordered exponents, Arch. Math., 84 (2005), p. 118-130 Article  MR 2120706 | Zbl 02155976

[3] Kaplansky, I. Maximal fields with valuations, Duke Math Journal, 9 (1942), p. 303-321 Article  MR 6161 | Zbl 0063.03135

[4] Kuhlmann, F.-V. Valuation theory of fields (Preprint)

[5] Leloup, G. Existentially equivalent cyclically ultrametric distances and cyclic valuations (2005) (submitted)

[6] Mac Lane, S. The uniqueness of the power series representation of certain fields with valuations, Annals of Mathematics, 39 (1938), p. 370-382 Article  MR 1503414 | Zbl 0019.04901

[7] Mac Lane, S. The universality of formal power series fields, Bulletin of the American Mathematical Society, 45 (1939), p. 888-890 Article  MR 610 | Zbl 0022.30401

[8] Neuman, B. H. On ordered division rings, Trans. Amer. Math. Soc., 66 (1949), p. 202-252 Article  MR 32593 | Zbl 0035.30401

[9] Redfield, R. H. Constructing lattice-ordered fields and division rings, Bull. Austral. Math. Soc., 40 (1989), p. 365-369 Article  MR 1037630 | Zbl 0683.12015

[10] Ribenboim, P. Théorie des Valuations, Les Presses de l’Université de Montréal, Montréal (1964) MR 249425 | Zbl 0139.26201