Course Identification

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.

This is a basic graduate course in algebra, whose purpose is to provide a good foundation for more advanced and specific 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).

Upon successful completion of the course students will be able to:

1. Demonstrate comprehension of the structure of modules over rings, in particular the structure of artinian and noetherian modules.
2. Apply the theorems and methods of representation theory and modules to obtain insights and understanding of the classical theorems in the structure of rings.
3. Demonstrate comprehension of the structure of non-commutative rings, in particular the structure of primitive and semi-primitive artinian rings.

• N. Jacobson, Basic Algebra II
• I. M. Isaacs, Algebra: A Graduate Course
• S. Lang, Algebra