Cyclically valued rings and formal power series

Leloup, Gérard

Leloup, Gérard

Annales mathématiques Blaise Pascal, Tome 14 (2007), p. 37-60 / Harvested from Numdam

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Résumé

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 $v (ϵ, \cdot)$ 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 $a = ϵ$ and the cyclically ordered group is indeed a totally ordered one.

Publié le : 2007-01-01

MR 2298803 | Zbl 1127.13019

DOI : https://doi.org/10.5802/ambp.226
Classification: 13F25, 13A18, 13A99, 06F15, 06F99

@article{AMBP_2007__14_1_37_0,
     author = {Leloup, G\'erard},
     title = {Cyclically valued rings and formal power series},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     year = {2007},
     pages = {37-60},
     doi = {10.5802/ambp.226},
     zbl = {1127.13019},
     mrnumber = {2298803},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2007__14_1_37_0}
}

Leloup, Gérard. Cyclically valued rings and formal power series. Annales mathématiques Blaise Pascal, Tome 14 (2007) pp. 37-60. doi : 10.5802/ambp.226. http://gdmltest.u-ga.fr/item/AMBP_2007__14_1_37_0/

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