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Course11110609839 lectures

Introduction to Commutative Algebra

Prof. A. V. Jayanthan · IIT Madras

Start with lecture 1 At NPTEL

Lectures

Ring theory basics

  1. Review of Ring Theory Video
  2. Review of Ring Theory (Ctd.) Video
  3. Ideals in commutative rings Video

Ideals

  1. Operations on ideals Video
  2. Properties of prime ideals Video
  3. Colon and Radical of ideals Video

Module theory basics

  1. Radicals, extension and contraction of ideals Video
  2. Modules and homomorphisms Video
  3. Isomorphism theorems and Operations on modules Video

Homomorphism and Nakayama’s Lemma

  1. Operations on modules (Ctd.) Video
  2. Module homomorphism and determinant trick Video
  3. Nakayama’s lemma and exact sequences Video

Properties of Modules

  1. Exact sequences (Ctd.) Video
  2. Homomorphisms and Tensor products Video
  3. Properties of tensor products Video

Localization

  1. Properties of tensor products (Ctd.) Video
  2. Tensor product of Algebras Video
  3. Localization Video
  4. Localization (Ctd.) Video

Localization and Integral dependence

  1. Local properties Video
  2. Further properties of localization Video
  3. Intergral dependence Video

Going-up Theorem

  1. Integral extensions Video
  2. Lying over and Going-up theorems Video
  3. Going-down theorem Video

Going-down theorem and Chain conditions

  1. Going-down theorem (Ctd.) Video
  2. Chain conditions Video
  3. Noetherian and Artinian modules Video
  4. Properties of Noetherian and Artinian modules, Composition Series Video

Properties of Noetherian rings

  1. Further properties of Noetherian and Artinian modules and rings Video
  2. Hilbert basis theorem and Primary decomposition Video
  3. Primary decomposition (Ctd.) Video

Primary decomposition and Artinian rings

  1. Uniqueness of primary decomposition Video
  2. 2nd Uniqueness theorem, Artinian rings Video
  3. Properties of Artinian rings Video

Noether Normalization and Hilbert’s Nullstellensatz

  1. Structure Theorem of Artinian rings Video
  2. Noether Normalization Video
  3. Hilberts Nullstellensatz Video

Live Session

  1. Live Session 02-03-2021 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Madras
Department
Mathematics
Teacher
Prof. A. V. Jayanthan
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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