The purpose is to begin to generalize the theory developed in [DM] (see below) to diagonal quadratic forms with non zero-divisor coefficients in reduced unitary pre-ordered commutative rings. We present:
1. Horn-geometric axioms entailing the equivalence between ring-theoretic isometry and representation by this class of forms is faithfully coded by the corresponding concepts in an associated reduced special group.
A preordered ring, <A, T>, satisying these axioms is called NT-faithfully quadratic (N for "multiplicative set of non zero-divisors in A").
2. A natural correspondence between rings that satisfy the axioms in (1) above and faithfully quadratic rings in the sense of [DM].
3. We prove that any reduced f-ring satisfies the axioms in (1) and, in fact, any preordered f-ring, <A,T>, where T is a preorder satisfying a certain weak cancelation property, verifies the axioms mentioned in item (1). In particular, this applies to real closed rings and rings of continuous functions (bounded or not) over any completely regular topological space.
Joint work with M. Dickmann (IMJ-PRG) and Hugo Ribeiro (USP).
[DM] M. Dickmann, F. Miraglia "Faithfully Quadratic Rings(Memoirs of the AMS 1128, november 2015).