The Resource Algebra : An Approach via Module Theory, by William A. Adkins, Steven H. Weintraub, (electronic resource)

Algebra : An Approach via Module Theory, by William A. Adkins, Steven H. Weintraub, (electronic resource)

Label
Algebra : An Approach via Module Theory
Title
Algebra
Title remainder
An Approach via Module Theory
Statement of responsibility
by William A. Adkins, Steven H. Weintraub
Creator
Contributor
Author
Subject
Language
eng
Summary
This book is designed as a text for a first-year graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generaliza­ tion of vector spaces. (To be precise, modules are to rings as vector spaces are to fields
Member of
http://library.link/vocab/creatorName
Adkins, William A
Dewey number
512
Image bit depth
0
LC call number
QA150-272
Literary form
non fiction
http://library.link/vocab/relatedWorkOrContributorName
  • Weintraub, Steven H.
  • SpringerLink (Online service)
Series statement
Graduate Texts in Mathematics
Series volume
136
http://library.link/vocab/subjectName
Algebra
Label
Algebra : An Approach via Module Theory, by William A. Adkins, Steven H. Weintraub, (electronic resource)
Link
https://doi.org/10.1007/978-1-4612-0923-2
Instantiates
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
1 Groups -- 1.1 Definitions and Examples -- 1.2 Subgroups and Cosets -- 1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups -- 1.4 Permutation Representations and the Sylow Theorems -- 1.5 The Symmetric Group and Symmetry Groups -- 1.6 Direct and Semidirect Products -- 1.7 Groups of Low Order -- 1.8 Exercises -- 2 Rings -- 2.1 Definitions and Examples -- 2.2 Ideals, Quotient Rings, and Isomorphism Theorems -- 2.3 Quotient Fields and Localization -- 2.4 Polynomial Rings -- 2.5 Principal Ideal Domains and Euclidean Domains -- 2.6 Unique Factorization Domains -- 2.7 Exercises -- 3 Modules and Vector Spaces -- 3.1 Definitions and Examples -- 3.2 Submodules and Quotient Modules -- 3.3 Direct Sums, Exact Sequences, and Horn -- 3.4 Free Modules -- 3.5 Projective Modules -- 3.6 Free Modules over a PID -- 3.7 Finitely Generated Modules over PIDs -- 3.8 Complemented Submodules -- 3.9 Exercises -- 4 Linear Algebra -- 4.1 Matrix Algebra -- 4.2 Determinants and Linear Equations -- 4.3 Matrix Representation of Homomorphisms -- 4.4 Canonical Form Theory -- 4.5 Computational Examples -- 4.6 Inner Product Spaces and Normal Linear Transformations -- 4.7 Exercises -- 5 Matrices over PIDs -- 5.1 Equivalence and Similarity -- 5.2 Hermite Normal Form -- 5.3 Smith Normal Form -- 5.4 Computational Examples -- 5.5 A Rank Criterion for Similarity -- 5.6 Exercises -- 6 Bilinear and Quadratic Forms -- 6.1 Duality -- 6.2 Bilinear and Sesquilinear Forms -- 6.3 Quadratic Forms -- 6.4 Exercises -- 7 Topics in Module Theory -- 7.1 Simple and Semisimple Rings and Modules -- 7.2 Multilinear Algebra -- 7.3 Exercises -- 8 Group Representations -- 8.1 Examples and General Results -- 8.2 Representations of Abelian Groups -- 8.3 Decomposition of the Regular Representation -- 8.4 Characters -- 8.5 Induced Representations -- 8.6 Permutation Representations -- 8.7 Concluding Remarks -- 8.8 Exercises -- Index of Notation -- Index of Terminology
Dimensions
unknown
Extent
1 online resource
File format
multiple file formats
Form of item
electronic
Isbn
9781461209232
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
c
Other control number
10.1007/978-1-4612-0923-2
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(DE-He213)978-1-4612-0923-2
Label
Algebra : An Approach via Module Theory, by William A. Adkins, Steven H. Weintraub, (electronic resource)
Link
https://doi.org/10.1007/978-1-4612-0923-2
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
1 Groups -- 1.1 Definitions and Examples -- 1.2 Subgroups and Cosets -- 1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups -- 1.4 Permutation Representations and the Sylow Theorems -- 1.5 The Symmetric Group and Symmetry Groups -- 1.6 Direct and Semidirect Products -- 1.7 Groups of Low Order -- 1.8 Exercises -- 2 Rings -- 2.1 Definitions and Examples -- 2.2 Ideals, Quotient Rings, and Isomorphism Theorems -- 2.3 Quotient Fields and Localization -- 2.4 Polynomial Rings -- 2.5 Principal Ideal Domains and Euclidean Domains -- 2.6 Unique Factorization Domains -- 2.7 Exercises -- 3 Modules and Vector Spaces -- 3.1 Definitions and Examples -- 3.2 Submodules and Quotient Modules -- 3.3 Direct Sums, Exact Sequences, and Horn -- 3.4 Free Modules -- 3.5 Projective Modules -- 3.6 Free Modules over a PID -- 3.7 Finitely Generated Modules over PIDs -- 3.8 Complemented Submodules -- 3.9 Exercises -- 4 Linear Algebra -- 4.1 Matrix Algebra -- 4.2 Determinants and Linear Equations -- 4.3 Matrix Representation of Homomorphisms -- 4.4 Canonical Form Theory -- 4.5 Computational Examples -- 4.6 Inner Product Spaces and Normal Linear Transformations -- 4.7 Exercises -- 5 Matrices over PIDs -- 5.1 Equivalence and Similarity -- 5.2 Hermite Normal Form -- 5.3 Smith Normal Form -- 5.4 Computational Examples -- 5.5 A Rank Criterion for Similarity -- 5.6 Exercises -- 6 Bilinear and Quadratic Forms -- 6.1 Duality -- 6.2 Bilinear and Sesquilinear Forms -- 6.3 Quadratic Forms -- 6.4 Exercises -- 7 Topics in Module Theory -- 7.1 Simple and Semisimple Rings and Modules -- 7.2 Multilinear Algebra -- 7.3 Exercises -- 8 Group Representations -- 8.1 Examples and General Results -- 8.2 Representations of Abelian Groups -- 8.3 Decomposition of the Regular Representation -- 8.4 Characters -- 8.5 Induced Representations -- 8.6 Permutation Representations -- 8.7 Concluding Remarks -- 8.8 Exercises -- Index of Notation -- Index of Terminology
Dimensions
unknown
Extent
1 online resource
File format
multiple file formats
Form of item
electronic
Isbn
9781461209232
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
c
Other control number
10.1007/978-1-4612-0923-2
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(DE-He213)978-1-4612-0923-2

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