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

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 zero is a sum of k +  1 products (resp. permuted products) of nth 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))) < ∞. (Take , R = ℝ (t ₁ ,…,t _k ) and ω(f(t ₁ ,…,t _k )) = f( − t _k ,t ₁  − t _k ,…,t _{k− 1}  − t _k ).)

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.

Keywords:

• Skew-fields
• Sums of products of nth powers

Mathematics Subject Classification:

• 12E15
• 16K40

Notes

Communicated by R. Parimala.