Syllabus for Math 611 -- Abstract Algebra

E. L. Lady

Part I. Modules over Rings (with emphasis on finiteness conditions) (Hungerford, Chapter 4).

  1. Examples of modules.
  2. Vector spaces (Hungerford 4.2).
  3. Length.
  4. Ascending & descending chain conditions (Hungerford 8.1).
  5. Krull-Schmidt-Azumaya Theorem.
  6. Exact sequences and commutative diagrams. (Hungerford 4.1).

Part II. Semi-simple Rings and Modules (Hungerford, Chapter 9).

  1. Zorn's Lemma (Hungerford, Introduction).
  2. Simple modules and semi-simple modules (Hungerford 9.3).
  3. Jacobson radical (Hungerford 9.2).
  4. Nakayama's Lemma (Hungerford 8.4).
  5. Wedderburn-Artin Theorem (Hungerford 9.3).
  6. Artinian rings are noetherian.

Part III. Commutative rings and modules over them (Hungerford, Chapter 8)

  1. Localization (Hungerford 3.4).
  2. Prime ideals. Krull dimension (Hungerford 8.2).
  3. Associated primes (Hungerford 8.3).
  4. Polynomial rings (Hungerford 3.5).
  5. Hilbert Basis Theorem (Hungerford 8.4).
  6. Division algorithm (Hungerford 3.6).
  7. Unique factorization domains (Hungerford 3.6).

Part IV. Categories, functors, and diagrams (Hungerford, Chapter 10)

  1. Categories. Monomorphisms & epimorphisms.
  2. Products & coproducts.
  3. Free modules (Hungerford 4.2).
  4. Projective and injective modules (Hungerford 4.3).
    EXCURSION: Injective modules over commutatative noetherian rings
  5. Left exact, right exact, and additive functors.
  6. Hom (Hungerford 4.4).
  7. Diagram chasing.
  8. Push-outs and pull-backs.
  9. Tensor product (Hungerford 4.5).
  10. Natural transformations.
  11. Adjoint functors (Hungerford 10.2).
  12. Flat modules.

Part V. Group actions (Hungerford, Sections 2.4, 2.5, 2.7).

  1. Isotropy subgroups and orbit (Hungerford 2.4).
  2. Sylow theorems (Hungerford 2.5).
  3. Nilpotent groups (Hungerford 2.7).
  4. Group rings.
  5. Group representations and characters.

Part VI. Modules over principal ideal domains & dedekind domains (Hungerford, Section 4.6).

  1. Characterization of dedekind domains.
  2. Torsion free implies flat.
  3. Structure of finitely generated modules (Hungerford 4.6).
  4. Pure submodules.

Part VII. Field Theory (Hungerford, Chapters 5 & 6).