Lectures related to Irreducible Factors and Noetherian Rings

Chinese Remainder Theorem: Rings and Fields

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

Principal Ideal Domains: Structure and Homomorphisms

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

Dedekind Rings: Theory and Applications

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

Commutative Algebra: Recollections

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

Irreducible Factors and Noetherian Rings

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

Congruence Relations in Rings

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

Discrete Valuation Rings

Explores discrete valuation rings, their properties, uniqueness of representation, and relationship with principal ideal domains.

Chinese Remainder Theorem: Euclidean Domains

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

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.

Properties of Euclidean Domains

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

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.

Factorisation in PIDs

Covers factorisation in PIDs, prime ideals, unique tuples, and common prime factors.

Finite Fields: Construction and Properties

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

Simple Modules: Schur's Lemma

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

Dedekind Rings: Integral Extensions and Noetherian Rings

Explores Dedekind rings, integral extensions, and noetherian rings in algebraic structures.

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.

Idempotent Elements and Central Orthogonal

Explores idempotent elements, central orthogonal elements, commutative rings, and prime ideals in non-central rings.

Primary Decomposition: Noetherian Ideal

Covers primary decomposition in Noetherian ideals and unique primary ideals.