Cyclically valued rings and formal power series

摘要

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 $k[[C,heta ]]$ 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 $k[[C,heta ]]$ with twisted multiplication is isomorphic to a $R^{prime}[[C^{prime},heta ^{prime}]]$, where $C^{prime}$ is a subgroup of the cyclically ordered group of all roots of $1$ in the field of complex numbers, and $R^{prime} simeq k[[H,heta ]]$, 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.