# Rings and Modules

- Institution: IIT Kharagpur
- Department: Mathematics
- Taught by: Prof. Mousumi Mandal, Prof. Ramakrishna Nanduri
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111105161)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111105161)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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

1. [Lecture-1 : Introduction to Rings](https://invisiblenetwork.ai/learn/nptel/111105161/fa895278689c) (YouTube)
2. [Lecture-2 : Rings, Subrings](https://invisiblenetwork.ai/learn/nptel/111105161/75c052de5793) (YouTube)
3. [Lecture-3 : Ring Homomorphism, Ideals](https://invisiblenetwork.ai/learn/nptel/111105161/afded956cce8) (YouTube)
4. [Lecture-4 : Properties of Ideals](https://invisiblenetwork.ai/learn/nptel/111105161/446abfc8f365) (YouTube)
5. [Lecture-5 : Properties of Ideals(contd.)](https://invisiblenetwork.ai/learn/nptel/111105161/2ead276cf319) (YouTube)
6. [Lecture-6 : Quotient Ring, Isomorphism Theorem](https://invisiblenetwork.ai/learn/nptel/111105161/61c6b8181230) (YouTube)
7. [Lecture-7 : Isomorphism Theorem, Homomorphism Theorem](https://invisiblenetwork.ai/learn/nptel/111105161/f64ead096109) (YouTube)
8. [Lecture-8 : Homomorphism Theorem](https://invisiblenetwork.ai/learn/nptel/111105161/d591ccf27072) (YouTube)
9. [Lecture-9 : Integral Domain, Quotient Ring](https://invisiblenetwork.ai/learn/nptel/111105161/8683e9e85d80) (YouTube)
10. [Lecture-10 : Quotient Ring](https://invisiblenetwork.ai/learn/nptel/111105161/16d57a86272f) (YouTube)
11. [Lecture-11 : Prime ideals, Maximal ideals](https://invisiblenetwork.ai/learn/nptel/111105161/23f6013a7e9f) (YouTube)
12. [Lecture-12 : Maximal ideals](https://invisiblenetwork.ai/learn/nptel/111105161/5f2442402e58) (YouTube)
13. [Lecture-13 : Hillbert’s Nullstellensatz](https://invisiblenetwork.ai/learn/nptel/111105161/6636e82cdb83) (YouTube)
14. [Lecture-14 : Hillbert’s Nullstellensatz (contd.)](https://invisiblenetwork.ai/learn/nptel/111105161/5ae59f0d8b6d) (YouTube)
15. [Lecture-15 : Application of Hillbert’s Nullstellensatz](https://invisiblenetwork.ai/learn/nptel/111105161/7e74e788342c) (YouTube)
16. [Lecture-16 : Unique Factorization domian](https://invisiblenetwork.ai/learn/nptel/111105161/142530b6d528) (YouTube)
17. [Lecture-17 : Properties of Unique Factorization domain](https://invisiblenetwork.ai/learn/nptel/111105161/aa14d3823955) (YouTube)
18. [Lecture-18 : Principal ideal domain](https://invisiblenetwork.ai/learn/nptel/111105161/a9273e90673d) (YouTube)
19. [Lecture-19 : Properties of PID and ED](https://invisiblenetwork.ai/learn/nptel/111105161/66320baa539a) (YouTube)
20. [Lecture-20 : Properties of PID and ED(contd.)](https://invisiblenetwork.ai/learn/nptel/111105161/0e52c4e7716e) (YouTube)
21. [Lecture-21 : Prime elements of Z\[i\]](https://invisiblenetwork.ai/learn/nptel/111105161/d6dcc6ea6e35) (YouTube)
22. [Lecture-22 : Prime elements of Z\[i\] (contd.)](https://invisiblenetwork.ai/learn/nptel/111105161/3bf3eb9c7918) (YouTube)
23. [Lecture-23 : Application in Z\[i\]](https://invisiblenetwork.ai/learn/nptel/111105161/e4d459d9ef75) (YouTube)
24. [Lecture-24 : Polynomial Rings over UFD](https://invisiblenetwork.ai/learn/nptel/111105161/054fd51949ab) (YouTube)
25. [Lecture-25 : Gauss's Lemma](https://invisiblenetwork.ai/learn/nptel/111105161/4b0c2ae5630e) (YouTube)
26. [Lecture-26: Polynomial Ring over UFD and Irreducibility Criterion](https://invisiblenetwork.ai/learn/nptel/111105161/f96f63657516) (YouTube)
27. [Lecture-27 : Irreducibility Criterion](https://invisiblenetwork.ai/learn/nptel/111105161/464e3d1ee3e6) (YouTube)
28. [Lecture-28 : Chinese Remainder Theorem](https://invisiblenetwork.ai/learn/nptel/111105161/3229025bf12f) (YouTube)
29. [Lecture-29 : Nilradical and Jacobson radical](https://invisiblenetwork.ai/learn/nptel/111105161/08072e2fdd1c) (YouTube)
30. [Lecture-30 : Examples and Problems](https://invisiblenetwork.ai/learn/nptel/111105161/f1ec0a980d11) (YouTube)
31. [Lecture-31 : Definition of Modules and Examples](https://invisiblenetwork.ai/learn/nptel/111105161/66fe878c5c90) (YouTube)
32. [Lecture-32 : Definition of Modules and Examples(contd.)](https://invisiblenetwork.ai/learn/nptel/111105161/3c11546bd69e) (YouTube)
33. [Lecture-33 : Submodules,direct sum and direct product of modules](https://invisiblenetwork.ai/learn/nptel/111105161/b9b8cd121064) (YouTube)
34. [Lecture-34 : Direct sum and direct product of modules, free modules](https://invisiblenetwork.ai/learn/nptel/111105161/09ef65634ea6) (YouTube)
35. [Lecture-35 : Finitely generated modules, free modules vs Vector spaces](https://invisiblenetwork.ai/learn/nptel/111105161/66b056c87f22) (YouTube)
36. [Lecture-36 : free modules vs Vector spaces](https://invisiblenetwork.ai/learn/nptel/111105161/60a227be3257) (YouTube)
37. [Lecture-37 : Vector spaces vs free modules & Examples](https://invisiblenetwork.ai/learn/nptel/111105161/d45d9fb6e136) (YouTube)
38. [Lecture-38 : Quotient modules and module homomorphisms](https://invisiblenetwork.ai/learn/nptel/111105161/0bfbac6182aa) (YouTube)
39. [Lecture-39 : Module homomorphism , Epimorphism theorem](https://invisiblenetwork.ai/learn/nptel/111105161/7f02e3d55d09) (YouTube)
40. [Lecture-40 : Epimorphism theorem](https://invisiblenetwork.ai/learn/nptel/111105161/0d4efffa5fdc) (YouTube)
41. [Lecture-41 : Maximal submodules, minimal submodules](https://invisiblenetwork.ai/learn/nptel/111105161/a446c80d8f20) (YouTube)
42. [Lecture-42 : Freeness of submodules of a free module over a PID](https://invisiblenetwork.ai/learn/nptel/111105161/655be5784a21) (YouTube)
43. [Lecture-43 : Torsion modules, freeness of torsion-free modules over a PID](https://invisiblenetwork.ai/learn/nptel/111105161/e2c4d39d1b7d) (YouTube)
44. [Lecture-44 : Rank of a module, p-submodules over a PID](https://invisiblenetwork.ai/learn/nptel/111105161/b62f5f38051b) (YouTube)
45. [Lecture-45 : Structure of a torsion module over a PID](https://invisiblenetwork.ai/learn/nptel/111105161/0c8c3891269d) (YouTube)
46. [Lecture-46 : Structure theorem, chain conditions](https://invisiblenetwork.ai/learn/nptel/111105161/ccf0276632d7) (YouTube)
47. [Lecture-47 : Artinian modules, Artinian rings](https://invisiblenetwork.ai/learn/nptel/111105161/d3887da5af3c) (YouTube)
48. [Lecture-48 : Noetherian modules, Noetherian rings](https://invisiblenetwork.ai/learn/nptel/111105161/2869021a411b) (YouTube)
49. [Lecture-49 : Ascending chain condition, Noetherian modules](https://invisiblenetwork.ai/learn/nptel/111105161/7f36be1cf25f) (YouTube)
50. [Lecture-50 : Examples of Noetherian and Artinian modules and rings](https://invisiblenetwork.ai/learn/nptel/111105161/c8e4b019a9d8) (YouTube)
51. [Lecture-51 : Composition series, Modules of finite length](https://invisiblenetwork.ai/learn/nptel/111105161/98bf15b2d409) (YouTube)
52. [Lecture-52 : Jordan-Holder’s theorem](https://invisiblenetwork.ai/learn/nptel/111105161/aa0f02907890) (YouTube)
53. [Lecture-53 : Artinian rings](https://invisiblenetwork.ai/learn/nptel/111105161/7484d6a33007) (YouTube)
54. [Lecture-54 : Noetherian rings](https://invisiblenetwork.ai/learn/nptel/111105161/6c6aebb0bee2) (YouTube)
55. [Lecture-55 : Hilbert basis theorem](https://invisiblenetwork.ai/learn/nptel/111105161/31b674785ad6) (YouTube)
56. [Lecture-56 : Cohen’s theorem on Noetherianness](https://invisiblenetwork.ai/learn/nptel/111105161/cc1f08e17394) (YouTube)
57. [Lecture-57 : Nakayama lemma](https://invisiblenetwork.ai/learn/nptel/111105161/cdc40cff4580) (YouTube)
58. [Lecture-58 : Nil and Jacobson radicals in Artinian rings](https://invisiblenetwork.ai/learn/nptel/111105161/237cf85fc231) (YouTube)
59. [Lecture-59 : Structure theorem](https://invisiblenetwork.ai/learn/nptel/111105161/74764c175419) (YouTube)
60. [Lecture-60 : Comparison between Artinian & Noetherian rings](https://invisiblenetwork.ai/learn/nptel/111105161/24669d637ca6) (YouTube)