## Syllabus for Math 611 -- Abstract Algebra

### E. L. Lady

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

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

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

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

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

- Localization (Hungerford 3.4).
- Prime ideals. Krull dimension (Hungerford 8.2).
- Associated primes (Hungerford 8.3).
- Polynomial rings (Hungerford 3.5).
- Hilbert Basis Theorem (Hungerford 8.4).
- Division algorithm (Hungerford 3.6).
- Unique factorization domains (Hungerford 3.6).

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

- Categories. Monomorphisms & epimorphisms.
- Products & coproducts.
- Free modules (Hungerford 4.2).
- Projective and injective modules (Hungerford 4.3).

EXCURSION:
Injective modules over commutatative noetherian rings
- Left exact, right exact, and additive functors.
- Hom (Hungerford 4.4).
- Diagram chasing.
- Push-outs and pull-backs.
- Tensor product (Hungerford 4.5).
- Natural transformations.
- Adjoint functors (Hungerford 10.2).
- Flat modules.

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

- Isotropy subgroups and orbit (Hungerford 2.4).
- Sylow theorems (Hungerford 2.5).
- Nilpotent groups (Hungerford 2.7).
- Group rings.
- Group representations and characters.

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

- Characterization of dedekind domains.
- Torsion free implies flat.
- Structure of finitely generated modules (Hungerford 4.6).
- Pure submodules.

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