Abstract Algebra

  1. Module Theory
    1. Introduction to Modules
      1. Definition of a Module over a Ring
        1. Left Modules
          1. Right Modules
          2. Modules vs. Vector Spaces
            1. Similarities and Differences
            2. Examples of Modules
              1. Vector Spaces as Modules over Fields
                1. Abelian Groups as Z-modules
                  1. Ideals as Modules
                2. Submodules and Homomorphisms
                  1. Submodules
                    1. Definition and Examples
                      1. Submodule Test
                      2. Quotient Modules
                        1. Construction and Properties
                          1. Well-Definedness
                          2. Module Homomorphisms
                            1. Definition and Properties
                              1. Kernel and Image
                              2. The Isomorphism Theorems for Modules
                                1. First Isomorphism Theorem
                                  1. Second Isomorphism Theorem
                                    1. Third Isomorphism Theorem
                                  2. Free Modules and Direct Sums
                                    1. Linear Combinations and Spanning Sets
                                      1. Definitions and Properties
                                      2. Linear Independence and Basis
                                        1. Definitions and Properties
                                        2. Free Modules
                                          1. Definition and Properties
                                            1. Existence of Bases
                                            2. Direct Sums of Modules
                                              1. Definition and Properties
                                                1. Internal Direct Sums
                                                2. Direct Products of Modules
                                                  1. Definition and Properties
                                                    1. Relationship to Direct Sums
                                                  2. Modules over a Principal Ideal Domain
                                                    1. Torsion Elements
                                                      1. Definition and Examples
                                                      2. Torsion Submodules
                                                        1. Structure and Properties
                                                        2. The Primary Decomposition Theorem
                                                          1. Statement and Applications
                                                          2. The Structure Theorem for Finitely Generated Modules over a PID
                                                            1. Statement and Consequences
                                                              1. Invariant Factor Form
                                                                1. Elementary Divisor Form
                                                              2. Applications to Linear Algebra
                                                                1. Similarity of Matrices
                                                                  1. Definition and Criteria
                                                                  2. Rational Canonical Form
                                                                    1. Construction and Properties
                                                                      1. Companion Matrices
                                                                      2. Jordan Canonical Form
                                                                        1. Construction and Properties
                                                                          1. Jordan Blocks

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                                                                      4. Field Theory and Galois Theory

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                                                                      1. Preliminaries and Foundational Concepts