This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. The course will cover the following topics (and perhaps some others).
Basic module theory and introductory homological algebra - most of Chapter 3 and part of Chapter 6 of Basic Algebra II: hom and tensor, projective and injective modules, abelian categories, resolutions, completely reducible modules, the Wedderburn-Artin theorem
Commutative ideal theory and Noetherian rings - part of Chapter 7 of Basic Algebra II: rings of polynomials, localization, primary decomposition theorem, Dedekind domains, Noether normalization
Galois Theory - Chapter 4 of Basic Algebra I and part of Chapter 8 of Basic Algebra II: algebraic and transcendental extensions, separable and normal extensions, the Galois group, solvability of equations by radicals
Jacobson, "Basic Algebra", Volumes 1 and 2, second edition.
Any standard course in abstract algebra for undergraduates and/or Math 551. It will be assumed that students understand the concepts of groups, rings, modules, vector space and linear algebra, and finitely generated modules over principal ideal domains.