Leloup, Gérard
Cyclically valued rings and formal power series
Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1 , p. 37-60
MR 2298803 | Zbl 1127.13019 | 1 citation dans Numdam
doi : 10.5802/ambp.226
URL stable : http://www.numdam.org/item?id=AMBP_2007__14_1_37_0

Classification:  13F25,  13A18,  13A99,  06F15,  06F99
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,·) which is closer to the usual valuations because, with the topology defined by v(a,·), 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.


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