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Abstract Algebra Refresher:
Review, Amplification, Examples
Thomas R. Shemanske
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Front Matter
Colophon
Preface
1
A quick review of a first course
1.1
What is Algebra?
1.2
Partitions and Equivalence Relations
1.3
Structure-preserving maps and quotient structures
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
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
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
4.1
Basic Definitions
Back Matter
References and Suggested Readings
Abstract Algebra Refresher:
Review, Amplification, Examples
Thomas R. Shemanske
Department of Mathematics
Dartmouth College
thomas.r.shemanske@dartmouth.edu
Version: January 19, 2025
Colophon
Preface
