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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)

Review of Ring Theory


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

References
Introduction to Commutative Algebra
Instructor: Prof. A. V. Jayanthan, Department of Mathematics, IIT Madras. This is an introductory course in Commutative Algebra.