Linear Algebra Over Division Ring PDF Download
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Author: Aleks Kleyn
Publisher: CreateSpace
ISBN: 9781499324006
Category : Mathematics
Languages : en
Pages : 108
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Book Description
In this book I treat linear maps of vector space over division ring. The set of linear maps of left vector space over division ring D is right vector space over division ring D. The concept of twin representations follows from the joint consideration of vector space V and vector space of linear transformations of the vector space V. Considering of twin representations of division ring in Abelian group leads to the concept of D-vector space and their linear map. Based on polylinear map I considered definition of tensor product of rings and tensor product of D-vector spaces.
Author: Aleks Kleyn
Publisher: CreateSpace
ISBN: 9781499324006
Category : Mathematics
Languages : en
Pages : 108
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Book Description
In this book I treat linear maps of vector space over division ring. The set of linear maps of left vector space over division ring D is right vector space over division ring D. The concept of twin representations follows from the joint consideration of vector space V and vector space of linear transformations of the vector space V. Considering of twin representations of division ring in Abelian group leads to the concept of D-vector space and their linear map. Based on polylinear map I considered definition of tensor product of rings and tensor product of D-vector spaces.
Author: Aleks Kleyn
Publisher: CreateSpace
ISBN: 9781502982476
Category :
Languages : ru
Pages : 68
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Book Description
In this book I treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a new picture. Matrices allow two products linked by transpose. Biring is algebra which defines on the set two correlated structures of the ring. As in the commutative case, solutions of a system of linear equations build up right or left vector space depending on type of system. We study vector spaces together with the system of linear equations because their properties have a close relationship. As in a commutative case, the group of automorphisms of a vector space has a single transitive representation on a basis manifold. This gives us an opportunity to introduce passive and active representations. Studying a vector space over a division ring uncovers new details in the relationship between passive and active transformations, makes this picture clearer.
Author: Bernard R. McDonald
Publisher: CRC Press
ISBN: 1000146464
Category : Mathematics
Languages : en
Pages : 563
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Book Description
This monograph arose from lectures at the University of Oklahoma on topics related to linear algebra over commutative rings. It provides an introduction of matrix theory over commutative rings. The monograph discusses the structure theory of a projective module.
Author: Matej Brešar
Publisher: Springer
ISBN: 3319086936
Category : Mathematics
Languages : en
Pages : 227
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Book Description
Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
Author: David J. Saltman
Publisher: American Mathematical Soc.
ISBN: 9780821889381
Category : Mathematics
Languages : en
Pages : 132
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Book Description
This volume is based on lectures on division algebras given at a conference held at Colorado State University. Although division algebras are a very classical object, this book presents this "classical" material in a new way, highlighting current approaches and new theorems, and illuminating the connections with a variety of areas in mathematics.
Author: M. Shirvani
Publisher: Cambridge University Press
ISBN: 0521339251
Category : Mathematics
Languages : en
Pages : 265
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Book Description
This book is concerned with subgroups of groups of the form GL(n,D) for some division ring D. In it the authors bring together many of the advances in the theory of skew linear groups. Some aspects of skew linear groups are similar to those for linear groups, however there are often significant differences either in the method of proof or the results themselves. Topics covered in this volume include irreducibility, unipotence, locally finite-dimensional division algebras, and division algebras associated with polycyclic groups. Both authors are experts in this area of current interest in group theory, and algebraists and research students will find this an accessible account of the subject.
Author: Alexander Abian
Publisher: Elsevier
ISBN: 1483186792
Category : Mathematics
Languages : en
Pages : 175
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Book Description
Linear Associative Algebras focuses on finite dimensional linear associative algebras and the Wedderburn structure theorems. The publication first elaborates on semigroups and groups, rings and fields, direct sum and tensor product of rings, and polynomial and matrix rings. The text then ponders on vector spaces, including finite dimensional vector spaces and matrix representation of vectors. The book takes a look at linear associative algebras, as well as the idempotent and nilpotent elements of an algebra, ideals of an algebra, total matrix algebras and the canonical forms of matrices, matrix representation of algebras, and division of algebras. The manuscript also tackles the Wedderburn structure theorems, including direct sum and tensor product decomposition of algebras, nilpotent algebras and the radical of an algebra, and structure of simple and semi-simple algebras. The publication is highly recommended for mathematicians and students interested in the Wedderburn structure theorems and finite dimensional linear associative algebras.
Author: Bernard R. McDonald
Publisher: CRC Press
ISBN: 1000110613
Category : Mathematics
Languages : en
Pages : 561
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Book Description
This monograph arose from lectures at the University of Oklahoma on topics related to linear algebra over commutative rings. It provides an introduction of matrix theory over commutative rings. The monograph discusses the structure theory of a projective module.
Author: Irving Kaplansky
Publisher: Courier Corporation
ISBN: 9780486432335
Category : Mathematics
Languages : en
Pages : 182
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Book Description
The author of this text seeks to remedy a common failing in teaching algebra: the neglect of related instruction in geometry. Focusing on inner product spaces, orthogonal similarity, and elements of geometry, this volume is illustrated with an abundance of examples, exercises, and proofs and is suitable for both undergraduate and graduate courses. 1974 edition.
Author: John Voight
Publisher: Springer Nature
ISBN: 3030566943
Category : Mathematics
Languages : en
Pages : 877
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Book Description
This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.
