Let A be a commutative unitary semi-real (– 1 is not a sum of squares) ring in which 2 is a unit ; let T = A^2 or a proper preorder of A. We shall describe first order axioms such that if the pair (A, T) is a model of these statements, then there is a special group, G_T(A), naturally associated to (A, T), faithfully coding the T-theory of diagonal quadratic forms with unit coefficients in A. Under mild restrictions, these axioms are necessary and sufficient for G_T(A) to faithfully code representation and T-isometry of non-singular diagonal quadratic forms with coefficients in A.

We shall present significant classes of rings satisfying these axioms and establish interesting results concerning the behavior of quadratic form theory over these classes of rings.

The needed concepts will be presented during the talk, which contains only part of the results published in :

M. Dickmann, F. Miraglia, Faithfully Quadratic Rings, Memoirs of the AMS, 1128, (November 2015).

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