This is a basic graduate course in algebra, whose purpose is to provide a
good foundation for more advanced and specific algebra courses. The course is suitable for first-year graduate students in mathematics and computer science who have studied at least 2 years of undergraduate algebra courses.
Topics to be covered in the course:
I. Structure theory of modules
1. Artinian and Noetherian modules
2. Schreier refinement theorem, Jordan-Holder theorem, Krull-Schmidt theorem.
3. Completely reducible modules, Schur?s lemma.
4. Tensor products of modules
5. Projective and injective modules
6. Wedderburn-Artin theorem for simple rings.
II. Structure theory of rings
1. Primitivity and semi-primitivity
2. Jacobson radical
3. Density theorems
4. Artinian rings
5. Wedderburn-Artin structure theorems for primitive and
semi-primitive artinian rings.
6. Commutative artinian rings and the Hilbert Nullstellensatz (if time
permits).
Bibliography:
N. Jacobson, Basic Algebra II
I. M. Isaacs, Algebra: A Graduate Course
S. Lang, Algebra