Lectures in course MATH-311: Algebra IV - rings and modules

Covers rings and modules, emphasizing color codes and homological algebra concepts.

Explores chain conditions in module theory, emphasizing Noetherian modules and stabilizing sequences of submodules.

Covers rings, modules, exact sequences, and finitely generated modules in the context of Noetherian and Artinian properties.

Introduces Noetherian modules and explores simple modules and composition series.

Covers the fundamental theorem of finitely generated modules over a PID.

Covers the Jordan Normal Form and the identification of torsion blocks within matrices.

Covers the basics of Homological algebra, focusing on Ext modules and their significance in modern mathematics.

Covers free resolution of modules and projective modules in module theory.

Covers homomorphisms, projective modules, and resolutions in chain complexes.

Explores chain homotopy, projective complexes, and homotopy equivalences in chain complexes.

Explores the long exact sequence of Ext-modules and their computations in homological algebra.

Covers exact sequences in homological algebra and the properties of Hom functors.

Explores the dimension theory of rings, focusing on chains of ideals and prime ideals.

Covers integral closures, focusing on Eisenstein primes and their properties.

Covers extending domains, leading coefficients, and valid extensions.

Covers the concept of tensor product in the context of bilinear maps and explores the uniqueness of tensor products.

Covers the concept of localization in algebra, focusing on making a ring multiplicative closed.

Covers the concept of localization and ideals in rings, focusing on extended and contracted ideals, integral extensions, and monic equations.

Covers the concept of prime ideals and local rings, emphasizing their importance.

Covers primary decomposition in commutative rings and its application in prime ideals.