Linear Equations Over a Commutative Ring PDF Download
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Author: Paul Camion
Publisher:
ISBN:
Category : Commutative rings
Languages : en
Pages : 33
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Book Description
Let A denote a n x m matrix over a commutative ring R and let x and l be n-rowed column vectors with elements in a module over R. Necessary and sufficient conditions are found that the equation Ax = l has a solution. (Author).
Author: Paul Camion
Publisher:
ISBN:
Category : Commutative rings
Languages : en
Pages : 33
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Book Description
Let A denote a n x m matrix over a commutative ring R and let x and l be n-rowed column vectors with elements in a module over R. Necessary and sufficient conditions are found that the equation Ax = l has a solution. (Author).
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: 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: Joachim Lambek
Publisher:
ISBN:
Category : Associative rings
Languages : en
Pages : 206
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Book Description
Author: K.P.S. Bhaskara Rao
Publisher: CRC Press
ISBN: 0203218876
Category : Mathematics
Languages : en
Pages : 192
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Book Description
The theory of generalized inverses of real or complex matrices has been expertly developed and documented. But the generalized inverses of matrices over rings have received comprehensive treatment only recently. In this book, the author, who contributed to the research and development of the theory, explains his results. He explores regular element
Author: James W. Brewer
Publisher: CRC Press
ISBN: 9780824775599
Category : Mathematics
Languages : en
Pages : 220
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Book Description
Author: Aleks Kleyn
Publisher: CreateSpace
ISBN: 9781477631812
Category : Mathematics
Languages : en
Pages : 60
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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 frame 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: Neal H. McCoy
Publisher: American Mathematical Soc.
ISBN: 1614440085
Category : Algebra, Abstract
Languages : en
Pages : 216
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Book Description
This monograph presents an introduction to that branch of abstract algebra having to do with the theory of rings, with some emphasis on the role of ideals in the theory. Except for a knowledge of certain fundamental theorems about determinants which is assumed in Chapter VIII, and at one point in Chapter VII, the book is almost entirely self-contained. Of course, the reader must have a certain amount of “mathematical maturity” in order to understand the illustrative examples and also to grasp the significance of the abstract approach. However, as far as formal technique is concerned, little more than the elements of algebra are presupposed.
Author: David Eisenbud
Publisher: Springer Science & Business Media
ISBN: 1461253500
Category : Mathematics
Languages : en
Pages : 784
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Book Description
This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included.
Author: Martin Kreuzer
Publisher: Springer
ISBN: 3319436015
Category : Mathematics
Languages : en
Pages : 332
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Book Description
This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.
