Cyclically valued rings and formal power series

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 -