Faithfully quadratic rings

Abstract

elds. Early on M. Knebusch observed that this set up might be used to investigate theories of quadratic forms over certain classes of (commutative, unitary) rings. This talk will present the main ideas and results of such investigation, initiated by the authors in 2004/2005, and now published in extenso in [2]. Our results deal with diagonal quadratic forms with invertible coecients over rings meeting some minimal conditions of orderability. The approach is based on an extension of the classical notion of isometry of quadratic forms to preordered rings (p-rings) (A;T ), where 2 is invertible, bringing the preorder T into play. We give three conditions in terms of this notion and the related notion of representation which are sucent |and, under mild restrictions, also necessary|to guarantee that the intrinsic theory of quadratic forms in (A;T ) coincides with the formal theory in the canonically associated (special) group GT (A). It follows that p-rings satisfying these axioms possess a rich theory of diagonal quadratic forms with invertible coecients, previously