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Introduction to Commutative Algebra

Introduction to Commutative Algebra. Instructor: Prof. A. V. Jayanthan, Department of Mathematics, IIT Madras. This is an introductory course in Commutative Algebra where most basic tools on commutative rings and modules over commutative rings are developed. This course is essential for anyone who wants to do research in areas such as commutative algebra, algebraic geometry, algebraic number theory etc. (from nptel.ac.in)

Lecture 01 - Review of Ring Theory

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Ring Theory Basics

Lecture 01 - Review of Ring Theory

Lecture 02 - Review of Ring Theory (cont.)

Lecture 03 - Ideals in Commutative Rings

Ideals

Lecture 04 - Operations on Ideals

Lecture 05 - Properties of Prime Ideals

Lecture 06 - Colon and Radical of Ideals

Module Theory Basics

Lecture 07 - Radicals, Extension and Contraction of Ideals

Lecture 08 - Modules and Homomorphisms

Lecture 09 - Isomorphism Theorems and Operations on Modules

Homomorphism and Nakayama's Lemma

Lecture 10 - Operations on Modules (cont.)

Lecture 11 - Module Homomorphism and Determinant Trick

Lecture 12 - Nakayama's Lemma and Exact Sequences

Properties of Modules

Lecture 13 - Exact Sequences (cont.)

Lecture 14 - Homomorphisms and Tensor Products

Lecture 15 - Properties of Tensor Products

Localization

Lecture 16 - Properties of Tensor Products (cont.)

Lecture 17 - Tensor Product of Algebras

Lecture 18 - Localization

Lecture 19 - Localization (cont.)

Localization and Integral Dependence

Lecture 20 - Local Properties

Lecture 21 - Further Properties of Localization

Lecture 22 - Integral Dependence

Going-up Theorem

Lecture 23 - Integral Extensions

Lecture 24 - Lying Over and Going-up Theorems

Lecture 25 - Going-down Theorem

Going-down Theorem and Chain Conditions

Lecture 26 - Going-down Theorem (cont.)

Lecture 27 - Chain Conditions

Lecture 28 - Noetherian and Artinian Modules

Lecture 29 - Properties of Noetherian and Artinian Modules, Composition Series

Properties of Noetherian Rings

Lecture 30 - Further Properties of Noetherian and Artinian Modules and Rings

Lecture 31 - Hilbert Basis Theorem and Primary Decomposition

Lecture 32 - Primary Decomposition (cont.)

Primary Decomposition and Artinian Rings

Lecture 33 - Uniqueness of Primary Decomposition

Lecture 34 - 2nd Uniqueness Theorem, Artinian Rings

Lecture 35 - Properties of Artinian Rings

Noether Normalization and Hilbert's Nullstellensatz

Lecture 36 - Structure Theorem of Artinian Rings

Lecture 37 - Noether Normalization

Lecture 38 - Hilbert's Nullstellensatz