Higher product levels of domains

The n-th product level of a skew–field D, psn(D), is a generalization of the n-th level of a field F, sn(F). An explicit bound for s2m(F) in terms of m and s2(F) is known and it is also known that there is no such bound for ps2m(D) when m is even. Our aim is to explicitly construct such a bound for odd m. More precisely, we construct a function image, such that ps2kl(D)less-than-or-equals, slantf(ps2k(D),k,l), for every integer k, every odd integer l and every skew–field D. We give an explicit bound for the n-th Pythagorean number of a simple extension image in Section 2 and prove a noncommutative version of Hilbert identities in Section 3. These two results are used in Section 4 in the proof of our main result. In section 5, we show that the n-th product level of an Ore domain is equal to the n-th product level of its skew field of fractions.