Cyclically valued rings and formal power series

Abstract

Rings of formal power series k[[C]] with exponents in a cyclically ordered group C were defined in [2]. Now, there exists a on k[[C]]: for every a in k[[C]] and c in C, we let v(c, a) be the first element of the support of a 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 υ(∈,·) which is closer to the usual valuations because, with the topology defined by υ(a,·), a cyclically valued ring is a topological ring if and only if a = ∈ and the cyclically ordered group is indeed a totally ordered one.