Cyclically valued rings and formal power series

Leloup, Gérard¹

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, Tome 14 (2007) no. 1, pp. 37-60

Résumé

Rings of formal power series $k\left[\right[C\left]\right]$ with exponents in a cyclically ordered group $C$ were defined in [2]. Now, there exists a “valuation” on $k\left[\right[C\left]\right]$ : for every $\sigma$ in $k\left[\right[C\left]\right]$ 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 $k\left[\right[C\left]\right]$. We prove that a cyclically valued ring is a subring of a power series ring $k\left[\right[C,\theta \left]\right]$ 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 $k\left[\right[C,\theta \left]\right]$ with twisted multiplication is isomorphic to a $R^{\prime }\left[[C^{\prime },{\theta }^{\prime }]\right]$, where $C^{\prime }$ is a subgroup of the cyclically ordered group of all roots of $1$ in the field of complex numbers, and $R^{\prime }\simeq k\left[[H,\theta ]\right]$, 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.

Détail

DOI : 10.5802/ambp.226

Classification : 13F25, 13A18, 13A99, 06F15, 06F99