| Summary: | In this monograph we extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where -1 is not a sum of squares and 2 is invertible. We accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of T-isometry, where T is a preorder of the given ring, A, or T = A². (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case. Under these axioms we prove that the ring-theoretic approach based on T-isometry coincides with the formal approach formulated in terms of reduced special groups. This guarantees, for rings verifying these axioms, the validity of a number of important structural properties, notably the Arason-Pfister Hauptsatz, Milnor's mod 2 Witt ring conjecture, Marshall's signature conjecture, uniform upper bounds for the Pfister index of quadratic forms, a local-global Sylvester inertia law, etc. We call (T)-faithfully quadratic rings verifying these axioms. A significant part of the monograph is devoted to prove quadratic faithfulness of certain outstanding (classes of) rings; among them, rings with many units satisfying a mild additional requirement, reduced f-rings (herein rings of continuous real-valued functions), and strictly representable rings. Obviously, T-quadratic faithfulness depends on both the ring and the preorder T. We isolate a property of preorders defined solely in terms of the real spectrum of a given ring -- that we baptise unit-reflecting preorders -- which, for an extensive class of preordered rings, (A, T), turns out to be equivalent to the T-quadratic faithfulness of A. We show, e.g., that all preorders on the ring of continuous real-valued functions on a compact Hausdorff are unit-reflecting; we also give examples where this property fails.
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