1. Modules

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;

Free modules;

Structure theorems, applications (e.g. structure of finitely generated abelian groups, reduction of endomorphisms).

3. Rings

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...