Leloup, Gérard
Cyclically valued rings and formal power series. Annales mathématiques Blaise Pascal, 14 no. 1 (2007), p. 37-60
Full text djvu | pdf | Reviews MR 2298803 | Zbl 1127.13019
Class. Math.: 13F25, 13A18, 13A99, 06F15, 06F99

stable URL: http://www.numdam.org/item?id=AMBP_2007__14_1_37_0

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.

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