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Faithfully Quadratic Rings


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English | ISBN: 1470414686 | 2015 | 148 Pages | PDF | 1 MB

In this monograph the authors 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. They 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 $T = A^2$. (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.
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