Learn · NPTEL

Course11110516160 lectures

Rings and Modules

Prof. Mousumi Mandal and Prof. Ramakrishna Nanduri · IIT Kharagpur

Start with lecture 1 At NPTEL

About

This course is a self-contained elementary introduction to Rings and Modules. We will cover basic topics of Ring Theory and Module Theory which is a core course in Algebra. This course is compulsory to those who want to pursue research in Algebra. This is a 12 weeks course. In the first six weeks we focus on the Ring Theory part and the last six weeks we focus on the Module Theory part.

Prerequisites: Group Theory, Linear AlgebraINTENDED-AUDIENCE: All learneres interested in Mathematics

Lectures

Week 1

  1. Lecture-1 : Introduction to Rings YouTube
  2. Lecture-2 : Rings, Subrings YouTube
  3. Lecture-3 : Ring Homomorphism, Ideals YouTube
  4. Lecture-4 : Properties of Ideals YouTube
  5. Lecture-5 : Properties of Ideals(contd.) YouTube

Week 2

  1. Lecture-6 : Quotient Ring, Isomorphism Theorem YouTube
  2. Lecture-7 : Isomorphism Theorem, Homomorphism Theorem YouTube
  3. Lecture-8 : Homomorphism Theorem YouTube
  4. Lecture-9 : Integral Domain, Quotient Ring YouTube
  5. Lecture-10 : Quotient Ring YouTube

Week 3

  1. Lecture-11 : Prime ideals, Maximal ideals YouTube
  2. Lecture-12 : Maximal ideals YouTube
  3. Lecture-13 : Hillbert’s Nullstellensatz YouTube
  4. Lecture-14 : Hillbert’s Nullstellensatz (contd.) YouTube
  5. Lecture-15 : Application of Hillbert’s Nullstellensatz YouTube

Week 4

  1. Lecture-16 : Unique Factorization domian YouTube
  2. Lecture-17 : Properties of Unique Factorization domain YouTube
  3. Lecture-18 : Principal ideal domain YouTube
  4. Lecture-19 : Properties of PID and ED YouTube
  5. Lecture-20 : Properties of PID and ED(contd.) YouTube

Week 5

  1. Lecture-21 : Prime elements of Z[i] YouTube
  2. Lecture-22 : Prime elements of Z[i] (contd.) YouTube
  3. Lecture-23 : Application in Z[i] YouTube
  4. Lecture-24 : Polynomial Rings over UFD YouTube
  5. Lecture-25 : Gauss's Lemma YouTube

Week 6

  1. Lecture-26: Polynomial Ring over UFD and Irreducibility Criterion YouTube
  2. Lecture-27 : Irreducibility Criterion YouTube
  3. Lecture-28 : Chinese Remainder Theorem YouTube
  4. Lecture-29 : Nilradical and Jacobson radical YouTube
  5. Lecture-30 : Examples and Problems YouTube

Week 7

  1. Lecture-31 : Definition of Modules and Examples YouTube
  2. Lecture-32 : Definition of Modules and Examples(contd.) YouTube
  3. Lecture-33 : Submodules,direct sum and direct product of modules YouTube
  4. Lecture-34 : Direct sum and direct product of modules, free modules YouTube
  5. Lecture-35 : Finitely generated modules, free modules vs Vector spaces YouTube

Week 8

  1. Lecture-36 : free modules vs Vector spaces YouTube
  2. Lecture-37 : Vector spaces vs free modules & Examples YouTube
  3. Lecture-38 : Quotient modules and module homomorphisms YouTube
  4. Lecture-39 : Module homomorphism , Epimorphism theorem YouTube
  5. Lecture-40 : Epimorphism theorem YouTube

Week 9

  1. Lecture-41 : Maximal submodules, minimal submodules YouTube
  2. Lecture-42 : Freeness of submodules of a free module over a PID YouTube
  3. Lecture-43 : Torsion modules, freeness of torsion-free modules over a PID YouTube
  4. Lecture-44 : Rank of a module, p-submodules over a PID YouTube
  5. Lecture-45 : Structure of a torsion module over a PID YouTube

Week 10

  1. Lecture-46 : Structure theorem, chain conditions YouTube
  2. Lecture-47 : Artinian modules, Artinian rings YouTube
  3. Lecture-48 : Noetherian modules, Noetherian rings YouTube
  4. Lecture-49 : Ascending chain condition, Noetherian modules YouTube
  5. Lecture-50 : Examples of Noetherian and Artinian modules and rings YouTube

Week 11

  1. Lecture-51 : Composition series, Modules of finite length YouTube
  2. Lecture-52 : Jordan-Holder’s theorem YouTube
  3. Lecture-53 : Artinian rings YouTube
  4. Lecture-54 : Noetherian rings YouTube
  5. Lecture-55 : Hilbert basis theorem YouTube

Week 12

  1. Lecture-56 : Cohen’s theorem on Noetherianness YouTube
  2. Lecture-57 : Nakayama lemma YouTube
  3. Lecture-58 : Nil and Jacobson radicals in Artinian rings YouTube
  4. Lecture-59 : Structure theorem YouTube
  5. Lecture-60 : Comparison between Artinian & Noetherian rings YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Kharagpur
Department
Mathematics
Taught by
Prof. Mousumi Mandal, Prof. Ramakrishna Nanduri
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.
Who can watch
NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed