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 $v(\u03f5,\xb7)$ which is closer to the usual valuations because, with the topology defined by $v(a,\xb7)$, a cyclically valued ring is a topological ring if and only if $a=\u03f5$ and the cyclically ordered group is indeed a totally ordered one.

@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}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {14}, number = {1}, year = {2007}, pages = {37-60}, doi = {10.5802/ambp.226}, mrnumber = {2298803}, zbl = {1127.13019}, language = {en}, url = {http://www.numdam.org/item/AMBP_2007__14_1_37_0} }

Leloup, Gérard. Cyclically valued rings and formal power series. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 37-60. doi : 10.5802/ambp.226. http://www.numdam.org/item/AMBP_2007__14_1_37_0/

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