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MATH 4101: Advanced algebra


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...



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