MATH 4101: Advanced algebra
Notion of a module and general linear algebra: submodule, quotient, factorization of a linear map by the quotient;
Free module, module of finite type;
Matrix of a linear map, Cayley–Hamilton theorem.
2. Finitely generated modules over principal ideal domains
Finitely generated modules over principal ideal domains;
Structure theorems, applications (e.g. structure of finitely generated abelian groups, reduction of endomorphisms).
Local rings, Nakayama's lemma;
Localization of a ring, of a module, of an ideal;
Ring extensions, integrality, finiteness, notion of an integrally closed ring.
4. Tensor product
Tensor product over a field; tensor product of modules;
tensor product and exact sequences, notion of flatness;
tensor product of algebras;
extension of scalars.
If time allows, additional topics may be addressed, such as: discrete valuation rings, Dedekind rings, Nullstellensatz, criteria for flatness...