Author: Paul Moritz Cohn
Publisher:
ISBN:
Category : Associative rings
Languages : en
Pages : 632

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Book Description

Author: Paul Moritz Cohn
Publisher:
ISBN:
Category :
Languages : en
Pages : 346

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Book Description

Author: Cohn
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Book Description

Author: Paul Moritz Cohn
Publisher: Cambridge University Press
ISBN: 0521432170
Category : Mathematics
Languages : en
Pages : 522

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Book Description
Non-commutative fields (also called skew fields or division rings) have not been studied as thoroughly as their commutative counterparts and most accounts have hitherto been confined to division algebras, that is skew fields finite-dimensional over their centre. Based on the author's LMS lecture note volume Skew Field Constructions, the present work offers a comprehensive account of skew fields. The axiomatic foundation and a precise description of the embedding problem are followed by an account of algebraic and topological construction methods, in particular, the author's general embedding theory is presented with full proofs, leading to the construction of skew fields. The powerful coproduct theorems of G. M. Bergman are proved here as well as the properties of the matrix reduction functor, a useful but little-known construction providing a source of examples and counter-examples. The construction and basic properties of existentially closed skew fields are given, leading to an example of a model class with an infinite forcing companion which is not axiomatizable. The treatment of equations over skew fields has been simplified and extended by the use of matrix methods, and the beginnings of non-commutative algebraic geometry are presented, with a precise account of the problems that need to be overcome for a satisfactory theory. A separate chapter describes valuations and orderings on skew fields, with a construction applicable to free fields. Numerous exercises test the reader's understanding, presenting further aspects and open problems in concise form, and notes and comments at the ends of chapters provide historical background.

Author: A. H. Schofield
Publisher: Cambridge University Press
ISBN: 0521278538
Category : Mathematics
Languages : en
Pages : 237

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A study of representations of rings over skew fields.

Author: P. M. Cohn
Publisher: Cambridge University Press
ISBN: 1139454994
Category : Mathematics
Languages : en
Pages : 21

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Book Description
Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.

Author: Viet Dung Nguyen
Publisher: American Mathematical Soc.
ISBN: 0821843702
Category : Mathematics
Languages : en
Pages : 377

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Book Description
The papers in this volume contain results in active research areas in the theory of rings and modules, including non commutative and commutative ring theory, module theory, representation theory, and coding theory.

Author: P. M. Cohn
Publisher: Cambridge University Press
ISBN: 9780521853378
Category : Mathematics
Languages : en
Pages : 594

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Book Description
Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention.

Author: Alberto Facchini
Publisher: World Scientific
ISBN: 9814478970
Category : Mathematics
Languages : en
Pages : 403

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Book Description
Surveying the most influential developments in the field, this proceedings reviews the latest research on algebras and their representations, commutative and non-commutative rings, modules, conformal algebras, and torsion theories.The volume collects stimulating discussions from world-renowned names including Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith.

Author: Robert Wisbauer
Publisher: Routledge
ISBN: 1351447343
Category : Mathematics
Languages : en
Pages : 622

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Book Description
This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.