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Positivity in *-algebras can be defined either algebraically, by quadratic modules, or analytically, by *-representations. By the induction procedure for *-representations, we can lift the analytical notion of positivity from a *-subalgebra to the entire *-algebra. The aim in this article is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the *-algebra is induced from its intersection with the *-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.

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)))

The aim of this paper is to introduce the notion of the nth product level ps n of an associative unital ring and study its properties. Our main results are that every Noetherian ring A with ps n (A) < ∞ for some n has ps nl (A) < ∞ for every odd number l (Theorem 8) and that for every even n there exists a skew-field D with ps n (D) = 1 and ps 2n (D) = ∞ (Theorem 9). This is in a sharp contrast with the commutative case. Namely, by Proposition 4.6 in [1], for every commutative unital ring R with ps 2 (R) < ∞ we have that ps n (R) < ∞ for every n .

(2000). Complete precones on noncommutative integral domains. Communications in Algebra: Vol. 28, No. 1, pp. 103-119.