Cyclically valued rings and formal power series

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

Annales mathématiques Blaise Pascal (2007)

  • Volume: 14, Issue: 1, page 37-60
  • ISSN: 1259-1734

Abstract

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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 σ in k [ [ C ] ] and c in C , we let v ( c , σ ) be the first element of the support of σ 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 , θ ] ] 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 [ [ C , θ ] ] with twisted multiplication is isomorphic to a R [ [ C , θ ] ] , where C is a subgroup of the cyclically ordered group of all roots of 1 in the field of complex numbers, and R k [ [ H , θ ] ] , with H a totally ordered group. We define a valuation v ( ϵ , · ) which is closer to the usual valuations because, with the topology defined by v ( a , · ) , 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.

How to cite

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Leloup, Gérard. "Cyclically valued rings and formal power series." Annales mathématiques Blaise Pascal 14.1 (2007): 37-60. <http://eudml.org/doc/10540>.

@article{Leloup2007,
abstract = {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 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 $k[[C,\theta ]]$ 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.},
affiliation = {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},
author = {Leloup, Gérard},
journal = {Annales mathématiques Blaise Pascal},
language = {eng},
month = {1},
number = {1},
pages = {37-60},
publisher = {Annales mathématiques Blaise Pascal},
title = {Cyclically valued rings and formal power series},
url = {http://eudml.org/doc/10540},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Leloup, Gérard
TI - Cyclically valued rings and formal power series
JO - Annales mathématiques Blaise Pascal
DA - 2007/1//
PB - Annales mathématiques Blaise Pascal
VL - 14
IS - 1
SP - 37
EP - 60
AB - 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 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 $k[[C,\theta ]]$ 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.
LA - eng
UR - http://eudml.org/doc/10540
ER -

References

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  1. L. Fuchs, Partially Ordered Algebraic Structures, (1963), Pergamon Press Zbl0137.02001
  2. M. Giraudet, F.-V. Kuhlmann, G. Leloup, Formal power series with cyclically ordered exponents, Arch. Math. 84 (2005), 118-130 Zbl1090.13015MR2120706
  3. I. Kaplansky, Maximal fields with valuations, Duke Math Journal 9 (1942), 303-321 Zbl0063.03135MR6161
  4. F.-V. Kuhlmann, Valuation theory of fields Zbl1024.12007
  5. G. Leloup, Existentially equivalent cyclically ultrametric distances and cyclic valuations, (2005) Zbl1216.03052
  6. S. Mac Lane, The uniqueness of the power series representation of certain fields with valuations, Annals of Mathematics 39 (1938), 370-382 Zbl0019.04901MR1503414
  7. S. Mac Lane, The universality of formal power series fields, Bulletin of the American Mathematical Society 45 (1939), 888-890 Zbl0022.30401MR610
  8. B. H. Neuman, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202-252 Zbl0035.30401MR32593
  9. R. H. Redfield, Constructing lattice-ordered fields and division rings, Bull. Austral. Math. Soc. 40 (1989), 365-369 Zbl0683.12015MR1037630
  10. P. Ribenboim, Théorie des Valuations, (1964), Les Presses de l’Université de Montréal, Montréal Zbl0139.26201MR249425

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