Syllabus Math 501

 

Math 501. Abstract Algebra II Lecture Syllabus

1. Modules over non-commutative rings.
Review of left and right ideals in rings. Submodules and quotient modules, homomorphisms. Direct sums and products. Limits and colimits.

2. Categories.
Examples of categories and functors. Natural transformations. Products and coproducts. Free objects in a category.

3. More module theory.
Free modules and the Invariant Basis Property. Exact sequences of modules. Left exactness of Hom. Projective and injective modules. Chain conditions and composition series. Hilbert's Basis Theorem.

4. Semisimple rings.
Semisimple rings and modules. Group rings and Maschke's theorem. Wedderburn's theorem on semisimple artinian rings. Application to representations of finite groups.

5. Tensor products and multilinear algebra.
Tensor products of modules. Mapping property of tensor products. Right exactness of tensor products. Flat modules. Exterior and symmetric products.

As time allows, either 6 or 7 below or another topic at instructor's choice.

6. Introduction to homological algebra.
Complexes and resolutions. Denitions of Ext and Tor. Long exact sequences for Ext and Tor.

7. Introduction to algebraic geometry.
Ane varieties, the Nullstellensatz, Zariski topology, irreducible varieties.

Note. Since Math 501 is no longer a comp course, instructors have some latitude in their choice of class material. However, the major topics in 1-5 above should be covered. Instructors may substitute for 6 and 7 another topic of their choice in the general area.

Books that could be used include "Abstract Algebra" by Dummitt and Foote, "Algebra" by Hungerford, and "Advanced Modern Algebra" by Rotman.

Revised and approved by GAC 4/24/12.