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Abstract Algebra Refresher:
Review, Amplification, Examples
Thomas R. Shemanske
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Front Matter
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Colophon
Preface
1
A quick review of a first course
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1.1
What is Algebra?
1.2
Partitions and Equivalence Relations
1.3
Structure-preserving maps and quotient structures
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1.3.1
Morphisms
1.3.2
Quotients
1.3.3
Cosets, partitions, and equivalence relations
1.3.4
Introducing an algebraic structure on the set of cosets.
1.4
A fundamental isomorphism theorem for groups, rings, vector spaces
1.5
New algebraic objects from old: products and sums
2
Basic results in group theory
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2.1
Cosets and some applications
2.2
Understanding quotients and further isomorphism theorems
2.3
Group Actions and applications
2.4
Some structure and classification theorems
2.5
The Symmetric Group
3
Basic results in ring theory
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3.1
Basic definitions and motivations
3.2
Factoring in integral domains
3.3
Ideals and quotients
3.4
Euclidean domains, PIDs, UFDs and all that jazz
3.5
Identifying irreducibles
3.6
Applications
4
Definitions
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4.1
Basic Definitions
Back Matter
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References and Suggested Readings
Chapter
4
Definitions
Here we accumulate basic definitions and examples from a standard first course in abstract algebra.
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4.1
Basic Definitions
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