Faithfully Quadratic Rings

ISBN-13: 9781470414689 / Angielski

M. Dickmann;F. Miraglia

Opis

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.

Spis treści

• Basic concepts
• Rings and special groups
• The notion of T-faithfully quadratic ring
• Some basic consequences
• Idempotents, Products and T-isometry
• First-order axioms for quadratic faithfulness
• Rings with many units
• Transversality of representation in p-rings with bounded inversion
• Reduced f-rings
• Strictly representable rings
• Quadratic form theory over faithfully quadratic rings
• Bibliography
• Index of symbols
• Subject index