Cyclically valued rings and formal power series
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 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: 10.5802/ambp.226
Classification: 13F25, 13A18, 13A99, 06F15, 06F99
Gérard Leloup 1

1 U.M.R. 7056 (Équipe de Logique, Paris VII) et Département de Mathématiques, Faculté des Sciences, université du Maine avenue Olivier Messiaen 72085 Le Mans Cedex 9, FRANCE
@article{AMBP_2007__14_1_37_0,
author = {G\'erard Leloup},
title = {Cyclically valued rings and formal power series},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {37--60},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {14},
number = {1},
year = {2007},
doi = {10.5802/ambp.226},
mrnumber = {2298803},
zbl = {1127.13019},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.226/}
}
TY  - JOUR
AU  - Gérard Leloup
TI  - Cyclically valued rings and formal power series
JO  - Annales mathématiques Blaise Pascal
PY  - 2007
SP  - 37
EP  - 60
VL  - 14
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.226/
UR  - https://www.ams.org/mathscinet-getitem?mr=2298803
UR  - https://zbmath.org/?q=an%3A1127.13019
UR  - https://doi.org/10.5802/ambp.226
DO  - 10.5802/ambp.226
LA  - en
ID  - AMBP_2007__14_1_37_0
ER  -
%0 Journal Article
%A Gérard Leloup
%T Cyclically valued rings and formal power series
%J Annales mathématiques Blaise Pascal
%D 2007
%P 37-60
%V 14
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.226
%R 10.5802/ambp.226
%G en
%F AMBP_2007__14_1_37_0
Gérard Leloup. Cyclically valued rings and formal power series. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 37-60. doi : 10.5802/ambp.226. https://ambp.centre-mersenne.org/articles/10.5802/ambp.226/

[1] L. Fuchs Partially Ordered Algebraic Structures, Pergamon Press, 1963 | Zbl

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

[3] I. Kaplansky Maximal fields with valuations, Duke Math Journal, Volume 9 (1942), pp. 303-321 | DOI | MR | Zbl

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

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

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

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

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

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

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

Cited by Sources: