Cyclically valued rings and formal power series
Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, p. 37-60
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.
DOI : https://doi.org/10.5802/ambp.226
Classification:  13F25,  13A18,  13A99,  06F15,  06F99
@article{AMBP_2007__14_1_37_0,
author = {Leloup, G\'erard},
title = {Cyclically valued rings and formal power series},
journal = {Annales math\'ematiques Blaise Pascal},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {14},
number = {1},
year = {2007},
pages = {37-60},
doi = {10.5802/ambp.226},
mrnumber = {2298803},
zbl = {1127.13019},
language = {en},
url = {http://www.numdam.org/item/AMBP_2007__14_1_37_0}
}

Leloup, Gérard. Cyclically valued rings and formal power series. Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 37-60. doi : 10.5802/ambp.226. http://www.numdam.org/item/AMBP_2007__14_1_37_0/

[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., Tome 84 (2005), pp. 118-130 | Article | MR 2120706 | Zbl 02155976

[3] Kaplansky, I. Maximal fields with valuations, Duke Math Journal, Tome 9 (1942), pp. 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, Tome 39 (1938), pp. 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, Tome 45 (1939), pp. 888-890 | Article | MR 610 | Zbl 0022.30401

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

[9] Redfield, R. H. Constructing lattice-ordered fields and division rings, Bull. Austral. Math. Soc., Tome 40 (1989), pp. 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