eBook ISBN:  9781470426293 
Product Code:  MEMO/238/1128.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470426293 
Product Code:  MEMO/238/1128.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 238; 2015; 129 ppMSC: Primary 11; 12; 03; 06; 46; 54;
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, wellknown to hold in the field case.

Table of Contents

Chapters

Preface

1. Basic Concepts

2. Rings and Special Groups

3. The Notion of TFaithfully Quadratic Ring. Some Basic Consequences

4. Idempotents, Products and Tisometry

5. FirstOrder Axioms for Quadratic Faithfulness

6. Rings with Many Units

7. Transversality of Representation in prings with Bounded Inversion

8. Reduced $f$Rings

9. Strictly Representable Rings

10. Quadratic Form Theory over Faithfully Quadratic Rings


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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, wellknown to hold in the field case.

Chapters

Preface

1. Basic Concepts

2. Rings and Special Groups

3. The Notion of TFaithfully Quadratic Ring. Some Basic Consequences

4. Idempotents, Products and Tisometry

5. FirstOrder Axioms for Quadratic Faithfulness

6. Rings with Many Units

7. Transversality of Representation in prings with Bounded Inversion

8. Reduced $f$Rings

9. Strictly Representable Rings

10. Quadratic Form Theory over Faithfully Quadratic Rings