Lectures related to Discrete Valuation Rings

Principal Ideal Domains: Structure and Homomorphisms

Covers the concepts of ideals, principal ideal domains, and ring homomorphisms.

Irreducible Factors and Noetherian Rings

Explores irreducible factors, Noetherian rings, ideal stability, and unique factorization in rings.

Chinese Remainder Theorem: Rings and Fields

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.

Commutative Algebra: Recollections

Covers fundamental concepts in commutative algebra, including rings, units, zero divisors, and local rings.

Congruence Relations in Rings

Explores congruence relations in rings, principal ideals, ring homomorphisms, and the characteristic of rings.

Dedekind Rings: Theory and Applications

Explores Dedekind rings, integral closure, factorization of ideals, and Gauss' Lemma.

Chinese Remainder Theorem: Euclidean Domains

Explores the Chinese Remainder Theorem for Euclidean domains and the properties of commutative rings and fields.

Irreducible Factors and Noetherian Rings

Discusses irreducible factors in rings and the properties of Noetherian rings.

Integral Domains: Factorisation and Noetherian Rings

Explores factorisation in Principal Ideal Domains and Noetherian rings, emphasizing the integral closure concept and the factorisation of ideals in Dedekind rings.

Simple Modules: Schur's Lemma

Covers simple modules, endomorphisms, and Schur's lemma in module theory.

Local Noetherian Rings

Covers local Noetherian rings, integrally closed domains, discrete valuations, and fraction fields.

Properties of Euclidean Domains

Explores the properties of Euclidean domains, including gcd, lcm, and the Chinese remainder theorem for polynomial rings.

Finite Fields: Construction and Properties

Explores the construction and properties of finite fields, including irreducible polynomials and the Chinese Remainder Theorem.

Rings and Fields: Principal Ideals and Ring Homomorphisms

Covers principal ideals, ring homomorphisms, and more in commutative rings and fields.

Algebra: Fundamental Theorem

Covers a general introduction and discusses algebra, emphasizing the importance of unique factorization in algebraic structures.

Factorisation in Principal Ideal Domains

Explores factorisation in Principal Ideal Domains and the properties of prime numbers.

Dimension theory of rings

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.

Ring Operations: Ideals and Classes

Covers the operations in rings, ideals, classes, and quotient rings.

Dimension Theory of Rings

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

Primary Decomposition: Noetherian Ideal

Covers primary decomposition in Noetherian ideals and unique primary ideals.