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Algebra Over a Field: A Comprehensive Exploration
Author: Dr. Evelyn Reed, Professor of Mathematics, University of California, Berkeley. Dr. Reed has published extensively on abstract algebra, with a particular focus on representation theory and the applications of algebra over a field in algebraic geometry.
Publisher: Springer Nature, a leading global publisher of scientific and scholarly literature, specializing in mathematics and related fields.
Editor: Dr. Arthur Chen, Associate Professor of Mathematics, Stanford University. Dr. Chen's expertise lies in commutative algebra and its applications to algebraic number theory.
Keywords: algebra over a field, field extension, vector space, module, polynomial ring, ideal, ring theory, abstract algebra, linear algebra
Summary: This article provides a comprehensive overview of algebra over a field, encompassing fundamental concepts, key methodologies, and advanced applications. We explore the structure of algebras, their relationship to vector spaces and modules, and delve into important examples such as polynomial algebras and group algebras. Various techniques for analyzing algebras, including the use of ideals, representations, and field extensions, are discussed.
1. Introduction to Algebra Over a Field
An algebra over a field is a fundamental concept in abstract algebra. It combines the structures of a field and a vector space, creating a rich mathematical object with numerous applications across various branches of mathematics and beyond. Formally, an algebra A over a field F is a vector space over F equipped with a bilinear multiplication operation: A x A → A, such that for all a, b, c ∈ A and λ ∈ F:
(a+b)c = ac + bc (right distributivity)
a(b+c) = ab + ac (left distributivity)
λ(ab) = (λa)b = a(λb) (scalar multiplication compatibility)
This definition elegantly blends the algebraic properties of a field with the linear structure of a vector space. Understanding this interplay is crucial to grasping the essence of algebra over a field.
2. Key Examples of Algebras Over a Field
Several significant examples illustrate the breadth and depth of the concept of an algebra over a field.
Polynomial Algebras: The set of polynomials in one or more variables with coefficients from a field F forms an algebra over F. Multiplication is defined as polynomial multiplication.
Matrix Algebras: The set of n x n matrices with entries from a field F forms an algebra over F. The matrix operations of addition and multiplication define the algebra structure.
Group Algebras: Given a group G and a field F, the group algebra F[G] is the set of formal linear combinations of elements of G with coefficients from F. Multiplication is defined by extending the group operation linearly.
Function Algebras: The set of continuous functions from a topological space to a field F often forms an algebra over F, with pointwise addition and multiplication as the operations.
3. Modules and Algebras Over a Field
Algebras over a field are closely related to modules. In fact, an algebra A over a field F can be viewed as a module over F with the additional structure of multiplication. This perspective allows us to utilize the powerful tools of module theory to study algebras. Understanding the interplay between the module structure and the multiplicative structure of an algebra is crucial for many investigations. For example, the concept of a simple module provides valuable insights into the representation theory of algebras.
4. Ideals and Quotient Algebras
Ideals play a central role in the study of algebras over a field. A two-sided ideal I of an algebra A is a subspace of A that is closed under multiplication by elements of A from both the left and the right. The quotient algebra A/I is formed by factoring out the ideal I. This construction is fundamental for understanding the structure of algebras and for classifying them. The study of ideals and quotient algebras is deeply connected to the theory of ring theory, further enriching the study of algebras over a field.
5. Field Extensions and Algebras
Field extensions provide another crucial perspective on algebras over a field. A field extension K/F can be viewed as an algebra over F. The structure of the extension, such as its degree and Galois group, provides valuable information about the algebra. Conversely, studying algebras over a field can offer insights into the structure of field extensions.
6. Representations of Algebras
Representation theory provides a powerful tool for studying algebras over a field. A representation of an algebra A over a field F is a homomorphism from A to the algebra of linear transformations of a vector space over F. Representations allow us to translate algebraic problems into linear algebraic problems, which are often easier to handle. The study of irreducible representations and their decomposition is essential for understanding the structure of algebras.
7. Applications of Algebra Over a Field
Algebras over a field have far-reaching applications in various areas of mathematics and beyond. In physics, they are used to model symmetries and describe physical systems. In computer science, they are used in cryptography and coding theory. In engineering, they are crucial for signal processing and control systems. Their applications are constantly expanding as the theoretical understanding deepens.
8. Advanced Topics in Algebra Over a Field
Advanced topics in the study of algebra over a field include the theory of Lie algebras, Hopf algebras, and the study of specific classes of algebras such as division algebras and simple algebras. These areas offer a rich landscape for further exploration and research.
Conclusion
The study of algebra over a field provides a rich and rewarding journey into the heart of abstract algebra. The interplay between the algebraic structure of a field and the linear structure of a vector space gives rise to a wide variety of mathematical objects and techniques. The concepts and methodologies discussed here provide a solid foundation for further exploration of this vibrant and crucial area of mathematics.
FAQs
1. What is the difference between an algebra over a field and a vector space? An algebra over a field is a vector space equipped with an additional binary operation (multiplication) that satisfies certain axioms. A vector space only has addition and scalar multiplication.
2. What is a simple algebra? A simple algebra is an algebra with no nontrivial two-sided ideals.
3. What is the significance of the Frobenius theorem? The Frobenius theorem classifies all finite-dimensional associative division algebras over the real numbers.
4. How are algebras over a field used in physics? They are used to represent symmetries, such as the rotation group in quantum mechanics.
5. What is a representation of an algebra? A representation is a homomorphism from the algebra to the algebra of linear transformations of a vector space.
6. What is a radical of an algebra? The radical is the largest nilpotent ideal of an algebra.
7. What are some examples of non-commutative algebras over a field? Matrix algebras and group algebras are examples of non-commutative algebras.
8. How are algebras over a field used in cryptography? They are used in the design of cryptographic systems and for encoding and decoding messages.
9. What are some open problems in the theory of algebras over a field? Many open problems relate to the classification of algebras and their representations, particularly in infinite dimensions.
Related Articles
1. Field Extensions: An exploration of the structure and properties of field extensions, their role in Galois theory, and their connection to algebra over a field.
2. Module Theory: A detailed study of modules over rings, including the concept of free modules, projective modules, injective modules, and their relation to algebra over a field.
3. Representation Theory of Finite Groups: Focuses on the representation theory of finite groups, a key application area for the theory of group algebras, which are algebras over a field.
4. Lie Algebras: An introduction to Lie algebras, their structure theory, and their connection to Lie groups and other areas of mathematics.
5. Hopf Algebras: A comprehensive overview of Hopf algebras, their properties, and their applications in various fields, including topology and physics.
6. Polynomial Rings and Ideals: A detailed discussion of polynomial rings, their ideal structure, and their connection to algebraic geometry and commutative algebra.
7. Galois Theory: An in-depth study of Galois theory, including its connection to field extensions and the solution of polynomial equations.
8. Commutative Algebra: An overview of commutative algebra, a vital area of algebra that heavily utilizes the concepts of algebra over a field.
9. Noncommutative Algebra: An exploration of the properties and applications of non-commutative algebras, including their importance in various areas of mathematics and physics.
| algebra over a field: An Introduction to Nonassociative Algebras Richard D. Schafer, 2017-11-15 Concise graduate-level introductory study presents some of the important ideas and results in the theory of nonassociative algebras. Places particular emphasis on alternative and (commutative) Jordan algebras. 1966 edition. |
| algebra over a field: Central Simple Algebras and Galois Cohomology Philippe Gille, Tamás Szamuely, 2017-08-10 The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results. |
| algebra over a field: Finite-Dimensional Division Algebras over Fields Nathan Jacobson, 2009-12-09 Here, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called multiplication algebras of Riemann matrices. Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field. |
| algebra over a field: Introduction To Commutative Algebra Michael F. Atiyah, I.G. MacDonald, 2018-03-09 First Published in 2018. This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization. |
| algebra over a field: Counterexamples in Topology Lynn Arthur Steen, J. Arthur Seebach, 2013-04-22 Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Numerous problems and exercises correlated with examples. 1978 edition. Bibliography. |
| algebra over a field: Finite Dimensional Algebras Yurj A. Drozd, Vladimir V. Kirichenko, 2012-12-06 This English edition has an additional chapter Elements of Homological Al gebra. Homological methods appear to be effective in many problems in the theory of algebras; we hope their inclusion makes this book more complete and self-contained as a textbook. We have also taken this occasion to correct several inaccuracies and errors in the original Russian edition. We should like to express our gratitude to V. Dlab who has not only metic ulously translated the text, but has also contributed by writing an Appendix devoted to a new important class of algebras, viz. quasi-hereditary algebras. Finally, we are indebted to the publishers, Springer-Verlag, for enabling this book to reach such a wide audience in the world of mathematical community. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface The theory of finite dimensional algebras is one of the oldest branches of modern algebra. Its origin is linked to the work of Hamilton who discovered the famous algebra of quaternions, and Cayley who developed matrix theory. Later finite dimensional algebras were studied by a large number of mathematicians including B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. At the end of the last century T. Molien and E. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non-semi simple algebras. |
| algebra over a field: Associative Algebras R.S. Pierce, 2012-12-06 For many people there is life after 40; for some mathematicians there is algebra after Galois theory. The objective ofthis book is to prove the latter thesis. It is written primarily for students who have assimilated substantial portions of a standard first year graduate algebra textbook, and who have enjoyed the experience. The material that is presented here should not be fatal if it is swallowed by persons who are not members of that group. The objects of our attention in this book are associative algebras, mostly the ones that are finite dimensional over a field. This subject is ideal for a textbook that will lead graduate students into a specialized field of research. The major theorems on associative algebras inc1ude some of the most splendid results of the great heros of algebra: Wedderbum, Artin, Noether, Hasse, Brauer, Albert, Jacobson, and many others. The process of refine ment and c1arification has brought the proof of the gems in this subject to a level that can be appreciated by students with only modest background. The subject is almost unique in the wide range of contacts that it makes with other parts of mathematics. The study of associative algebras con tributes to and draws from such topics as group theory, commutative ring theory, field theory, algebraic number theory, algebraic geometry, homo logical algebra, and category theory. It even has some ties with parts of applied mathematics. |
| algebra over a field: Commutative Algebra Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson, 2010-09-29 Commutative algebra is a rapidly growing subject that is developing in many different directions. This volume presents several of the most recent results from various areas related to both Noetherian and non-Noetherian commutative algebra. This volume contains a collection of invited survey articles by some of the leading experts in the field. The authors of these chapters have been carefully selected for their important contributions to an area of commutative-algebraic research. Some topics presented in the volume include: generalizations of cyclic modules, zero divisor graphs, class semigroups, forcing algebras, syzygy bundles, tight closure, Gorenstein dimensions, tensor products of algebras over fields, as well as many others. This book is intended for researchers and graduate students interested in studying the many topics related to commutative algebra. |
| algebra over a field: Linear Algebra Over Division Ring Aleks Kleyn, 2014-10-27 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. |
| algebra over a field: Advanced Linear Algebra Hugo Woerdeman, 2015-12-23 Advanced Linear Algebra features a student-friendly approach to the theory of linear algebra. The author’s emphasis on vector spaces over general fields, with corresponding current applications, sets the book apart. He focuses on finite fields and complex numbers, and discusses matrix algebra over these fields. The text then proceeds to cover vector spaces in depth. Also discussed are standard topics in linear algebra including linear transformations, Jordan canonical form, inner product spaces, spectral theory, and, as supplementary topics, dual spaces, quotient spaces, and tensor products. Written in clear and concise language, the text sticks to the development of linear algebra without excessively addressing applications. A unique chapter on How to Use Linear Algebra is offered after the theory is presented. In addition, students are given pointers on how to start a research project. The proofs are clear and complete and the exercises are well designed. In addition, full solutions are included for almost all exercises. |
| algebra over a field: Algebras and Representation Theory Karin Erdmann, Thorsten Holm, 2018-09-07 This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers. The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel's celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams. Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses. |
| algebra over a field: Steps in Commutative Algebra R. Y. Sharp, 2000 Introductory account of commutative algebra, aimed at students with a background in basic algebra. |
| algebra over a field: (Mostly) Commutative Algebra Antoine Chambert-Loir, 2021-04-08 This book stems from lectures on commutative algebra for 4th-year university students at two French universities (Paris and Rennes). At that level, students have already followed a basic course in linear algebra and are essentially fluent with the language of vector spaces over fields. The topics introduced include arithmetic of rings, modules, especially principal ideal rings and the classification of modules over such rings, Galois theory, as well as an introduction to more advanced topics such as homological algebra, tensor products, and algebraic concepts involved in algebraic geometry. More than 300 exercises will allow the reader to deepen his understanding of the subject. The book also includes 11 historical vignettes about mathematicians who contributed to commutative algebra. |
| algebra over a field: Thirty-three Miniatures Jiří Matoušek, 2010 This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lovasz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for seminar presentations. Table of Contents: Fibonacci numbers, quickly; Fibonacci numbers, the formula; The clubs of Oddtown; Same-size intersections; Error-correcting codes; Odd distances; Are these distances Euclidean?; Packing complete bipartite graphs; Equiangular lines; Where is the triangle?; Checking matrix multiplication; Tiling a rectangle by squares; Three Petersens are not enough; Petersen, Hoffman-Singleton, and maybe 57; Only two distances; Covering a cube minus one vertex; Medium-size intersection is hard to avoid; On the difficulty of reducing the diameter; The end of the small coins; Walking in the yard; Counting spanning trees; In how many ways can a man tile a board?; More bricks--more walls?; Perfect matchings and determinants; Turning a ladder over a finite field; Counting compositions; Is it associative?; The secret agent and umbrella; Shannon capacity of the union: a tale of two fields; Equilateral sets; Cutting cheaply using eigenvectors; Rotating the cube; Set pairs and exterior products; Index. (STML/53) |
| algebra over a field: Quaternion Algebras John Voight, 2021-06-28 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. |
| algebra over a field: Algebra in Action: A Course in Groups, Rings, and Fields Shahriar Shahriar, 2017-08-16 This text—based on the author's popular courses at Pomona College—provides a readable, student-friendly, and somewhat sophisticated introduction to abstract algebra. It is aimed at sophomore or junior undergraduates who are seeing the material for the first time. In addition to the usual definitions and theorems, there is ample discussion to help students build intuition and learn how to think about the abstract concepts. The book has over 1300 exercises and mini-projects of varying degrees of difficulty, and, to facilitate active learning and self-study, hints and short answers for many of the problems are provided. There are full solutions to over 100 problems in order to augment the text and to model the writing of solutions. Lattice diagrams are used throughout to visually demonstrate results and proof techniques. The book covers groups, rings, and fields. In group theory, group actions are the unifying theme and are introduced early. Ring theory is motivated by what is needed for solving Diophantine equations, and, in field theory, Galois theory and the solvability of polynomials take center stage. In each area, the text goes deep enough to demonstrate the power of abstract thinking and to convince the reader that the subject is full of unexpected results. |
| algebra over a field: Algebraic Curves over a Finite Field J. W. P. Hirschfeld, Gabor Korchmaros, Fernando Torres, 2013-03-25 This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students. |
| algebra over a field: A Book of Abstract Algebra Charles C Pinter, 2010-01-14 Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition. |
| algebra over a field: The Linear Algebra a Beginning Graduate Student Ought to Know Jonathan S. Golan, 2007-04-05 This book rigorously deals with the abstract theory and, at the same time, devotes considerable space to the numerical and computational aspects of linear algebra. It features a large number of thumbnail portraits of researchers who have contributed to the development of linear algebra as we know it today and also includes over 1,000 exercises, many of which are very challenging. The book can be used as a self-study guide; a textbook for a course in advanced linear algebra, either at the upper-class undergraduate level or at the first-year graduate level; or as a reference book. |
| algebra over a field: A Singular Introduction to Commutative Algebra Gert-Martin Greuel, Gerhard Pfister, 2012-12-06 This book can be understood as a model for teaching commutative algebra, and takes into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, the authors show how the concept can be worked on using a computer. The computations are exemplified with the computer algebra system Singular, developed by the authors. Singular is a special system for polynomial computation with many features for global as well as for local commutative algebra and algebraic geometry. The book includes a CD containing Singular as well as the examples and procedures explained in the book. |
| algebra over a field: Algebraic Groups J. S. Milne, 2017-09-21 Comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, with few prerequisites. |
| algebra over a field: An Introduction to Algebraic Structures Joseph Landin, 2012-08-29 This self-contained text covers sets and numbers, elements of set theory, real numbers, the theory of groups, group isomorphism and homomorphism, theory of rings, and polynomial rings. 1969 edition. |
| algebra over a field: Lectures on Algebra Shreeram Shankar Abhyankar, 2006 This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel. Lectures on Algebra II, will serve as a textbook for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both books are based on the author's lectures at Purdue University over the last few years. |
| algebra over a field: Approximations and Endomorphism Algebras of Modules Rüdiger Göbel, Jan Trlifaj, 2012-10-01 This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory. |
| algebra over a field: Introduction to Applied Linear Algebra Stephen Boyd, Lieven Vandenberghe, 2018-06-07 A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples. |
| algebra over a field: Basic Abstract Algebra Robert B. Ash, 2013-06-17 Relations between groups and sets, results and methods of abstract algebra in terms of number theory and geometry, and noncommutative and homological algebra. Solutions. 2006 edition. |
| algebra over a field: Factorization Algebras in Quantum Field Theory Kevin Costello, Owen Gwilliam, 2017 This first volume develops factorization algebras with a focus upon examples exhibiting their use in field theory, which will be useful for researchers and graduates. |
| algebra over a field: Advances in Algebra and Model Theory M Droste, R. Gobel, 1998-01-29 Contains 25 surveys in algebra and model theory, all written by leading experts in the field. The surveys are based around talks given at conferences held in Essen, 1994, and Dresden, 1995. Each contribution is written in such a way as to highlight the ideas that were discussed at the conferences, and also to stimulate open research problems in a form accessible to the whole mathematical community. The topics include field and ring theory as well as groups, ordered algebraic structure and their relationship to model theory. Several papers deal with infinite permutation groups, abelian groups, modules and their relatives and representations. Model theoretic aspects include quantifier elimination in skew fields, Hilbert's 17th problem, (aleph-0)-categorical structures and Boolean algebras. Moreover symmetry questions and automorphism groups of orders are covered. This work contains 25 surveys in algebra and model theory, each is written in such a way as to highlight the ideas that were discussed at Conferences, and also to stimulate open research problems in a form accessible to the whole mathematical community. |
| algebra over a field: Structure of Algebras Abraham Adrian Albert, 1939-12-31 The first three chapters of this work contain an exposition of the Wedderburn structure theorems. Chapter IV contains the theory of the commutator subalgebra of a simple subalgebra of a normal simple algebra, the study of automorphisms of a simple algebra, splitting fields, and the index reduction factor theory. The fifth chapter contains the foundation of the theory of crossed products and of their special case, cyclic algebras. The theory of exponents is derived there as well as the consequent factorization of normal division algebras into direct factors of prime-power degree. Chapter VI consists of the study of the abelian group of cyclic systems which is applied in Chapter VII to yield the theory of the structure of direct products of cyclic algebras and the consequent properties of norms in cyclic fields. This chapter is closed with the theory of $p$-algebras. In Chapter VIII an exposition is given of the theory of the representations of algebras. The treatment is somewhat novel in that while the recent expositions have used representation theorems to obtain a number of results on algebras, here the theorems on algebras are themselves used in the derivation of results on representations. The presentation has its inspiration in the author's work on the theory of Riemann matrices and is concluded by the introduction to the generalization (by H. Weyl and the author) of that theory. The theory of involutorial simple algebras is derived in Chapter X both for algebras over general fields and over the rational field. The results are also applied in the determination of the structure of the multiplication algebras of all generalized Riemann matrices, a result which is seen in Chapter XI to imply a complete solution of the principal problem on Riemann matrices. |
| algebra over a field: Introduction to Non-linear Algebra Valeri? Valer?evich Dolotin, A. Morozov, Al?bert Dmitrievich Morozov, 2007 Literaturverz. S. 267 - 269 |
| algebra over a field: Algebraic Geometry and Commutative Algebra Hiroaki Hijikata, Heisuke Hironaka, Masaki Maruyama, 2014-05-10 Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata presents a collection of papers on algebraic geometry and commutative algebra in honor of Masayoshi Nagata for his significant contributions to commutative algebra. Topics covered range from power series rings and rings of invariants of finite linear groups to the convolution algebra of distributions on totally disconnected locally compact groups. The discussion begins with a description of several formulas for enumerating certain types of objects, which may be tabular arrangements of integers called Young tableaux or some types of monomials. The next chapter explains how to establish these enumerative formulas, with emphasis on the role played by transformations of determinantal polynomials and recurrence relations satisfied by them. The book then turns to several applications of the enumerative formulas and universal identity, including including enumerative proofs of the straightening law of Doubilet-Rota-Stein and computations of Hilbert functions of polynomial ideals of certain determinantal loci. Invariant differentials and quaternion extensions are also examined, along with the moduli of Todorov surfaces and the classification problem of embedded lines in characteristic p. This monograph will be a useful resource for practitioners and researchers in algebra and geometry. |
| algebra over a field: Rings, Fields and Groups R. B. J. T. Allenby, 1991 Provides an introduction to the results, methods and ideas which are now commonly studied in abstract algebra courses |
| algebra over a field: Introduction to Representation Theory Pavel I. Etingof, Oleg Golberg, Sebastian Hensel , Tiankai Liu , Alex Schwendner , Dmitry Vaintrob , Elena Yudovina , 2011 Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra. |
| algebra over a field: Collins Dictionary of Mathematics Ephraim J. Borowski, Jonathan M. Borwein, 2005 A fully revised and expanded edition of the popular (over 50,000 copies sold) and authoritative Collins Dictionary of Mathematics. |
| algebra over a field: Official Summary of Security Transactions and Holdings Reported to the Securities and Exchange Commission Under the Securities Exchange Act of 1934 and the Public Utility Holding Company Act of 1935 United States. Securities and Exchange Commission, 1988 |
| algebra over a field: Basic Algebra P.M. Cohn, 2012-12-06 This is the first volume of a revised edition of P.M. Cohn's classic three-volume text Algebra, widely regarded as one of the most outstanding introductory algebra textbooks. This volume covers the important results of algebra. Readers should have some knowledge of linear algebra, groups and fields, although all the essential facts and definitions are recalled. |
| algebra over a field: Algebraic Operads Jean-Louis Loday, Bruno Vallette, 2012-08-08 In many areas of mathematics some “higher operations” are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After a low-level chapter on Algebra, accessible to (advanced) undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices review the basic results needed in order to understand the various chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers. |
| algebra over a field: Invitation to Nonlinear Algebra Mateusz Michałek, Bernd Sturmfels, 2021-03-05 Nonlinear algebra provides modern mathematical tools to address challenges arising in the sciences and engineering. It is useful everywhere, where polynomials appear: in particular, data and computational sciences, statistics, physics, optimization. The book offers an invitation to this broad and fast-developing area. It is not an extensive encyclopedia of known results, but rather a first introduction to the subject, allowing the reader to enter into more advanced topics. It was designed as the next step after linear algebra and well before abstract algebraic geometry. The book presents both classical topics—like the Nullstellensatz and primary decomposition—and more modern ones—like tropical geometry and semidefinite programming. The focus lies on interactions and applications. Each of the thirteen chapters introduces fundamental concepts. The book may be used for a one-semester course, and the over 200 exercises will help the readers to deepen their understanding of the subject. |
| algebra over a field: Representation Theory Alexander Zimmermann, 2014-08-15 Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use. |
| algebra over a field: Introduction to Modern Algebra and Matrix Theory Otto Schreier, Emanuel Sperner, 2011-01-01 This unique text provides students with a basic course in both calculus and analytic geometry. It promotes an intuitive approach to calculus and emphasizes algebraic concepts. Minimal prerequisites. Numerous exercises. 1951 edition-- |
Algebra - Wikipedia
Elementary algebra, also called school algebra, college algebra, and classical algebra, [22] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on …
Introduction to Algebra - Math is Fun
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6". But instead of saying " obviously x=6", use this neat step-by-step …
Algebra I - Khan Academy
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a …
Algebra | History, Definition, & Facts | Britannica
May 9, 2025 · Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b - 2 = 5 are …
Algebra - What is Algebra? | Basic Algebra | Definition - Cuemath
Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, …
How to Understand Algebra (with Pictures) - wikiHow
Mar 18, 2025 · Algebra is a system of manipulating numbers and operations to try to solve problems. When you learn algebra, you will learn the rules to follow for solving problems. But to …
What is Algebra? - BYJU'S
Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the …
Algebra in Math - Definition, Branches, Basics and Examples
Apr 7, 2025 · This section covers key algebra concepts, including expressions, equations, operations, and methods for solving linear and quadratic equations, along with polynomials and …
Algebra - Simple English Wikipedia, the free encyclopedia
People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations …
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Free algebra tutorial and help. Notes, videos, steps. Solve and simplify linear, quadratic, polynomial, and rational expressions and equations.
Algebra - Wikipedia
Elementary algebra, also called school algebra, college algebra, and classical algebra, [22] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on …
Introduction to Algebra - Math is Fun
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6". But instead of saying " obviously x=6", use this neat step-by-step …
Algebra I - Khan Academy
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a …
Algebra | History, Definition, & Facts | Britannica
May 9, 2025 · Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b - 2 = 5 are …
Algebra - What is Algebra? | Basic Algebra | Definition - Cuemath
Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, …
How to Understand Algebra (with Pictures) - wikiHow
Mar 18, 2025 · Algebra is a system of manipulating numbers and operations to try to solve problems. When you learn algebra, you will learn the rules to follow for solving problems. But to …
What is Algebra? - BYJU'S
Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the …
Algebra in Math - Definition, Branches, Basics and Examples
Apr 7, 2025 · This section covers key algebra concepts, including expressions, equations, operations, and methods for solving linear and quadratic equations, along with polynomials and …
Algebra - Simple English Wikipedia, the free encyclopedia
People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations …
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Free algebra tutorial and help. Notes, videos, steps. Solve and simplify linear, quadratic, polynomial, and rational expressions and equations.
