A Variant of Higher Product Levels of Integral Domains and *-Domains

Abstract

Abstract. For every domain R and every even integer n we define ms n (R) (resp. ps n (R)) as the smallest number k such that 0 is a sum of k + 1 products (resp. permuted products) of n-th powers of nonzero elements from R. There are many results about ps n in the literature but nothing is known about ms n .We prove two results about ms n of twisted Laurent series rings R ((x, ω)). The first result is that if ms 2 (R) = ∞ and ω has order n/2 in Aut(R), then ms n (R((x, ω))) = ∞. The second result is that there exist R and ω such that ms n (R((x, ω))) = ∞ and ps n (R ((x, ω)Finally, we define ms n and ps n of a domain R with involution. For a certain involution on R ((x, ω)), we prove analogues of the first and the second result.

Generalized orderings and rings of fractions

“…In [Ci3] the first author constructed a skew-field D n such that −1 ∈ ΣP n (D n ) but −1 ∈ ΣΠ n (D n ). Clearly, ΣP n (D n ) satisfies the assertion (ii) of Corollary 5, hence it is a G 1 -cone for a nonabelian n-torsion group G 1 .…”