HIGHER PRODUCT LEVELS OF NONCOMMUTATIVE RINGS

Abstract

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 .

Keywords:

• Quadratic forms
• Noetherian rings
• Levels of rings
• Ordered rings

Acknowledgments