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 . 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
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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/

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