NOC:Rings and Modules

Lecture 1 - Introduction to Rings

Lecture 2 - Rings, Subrings

Lecture 3 - Ring Homomorphism, Ideals

Lecture 4 - Properties of Ideals

Lecture 5 - Properties of Ideals (Continued...)

Lecture 6 - Quotient Ring, Isomorphism Theorem

Lecture 7 - Isomorphism Theorem, Homomorphism Theorem

Lecture 8 - Homomorphism Theorem

Lecture 9 - Integral Domain, Quotient Ring

Lecture 10 - Quotient Ring

Lecture 11 - Prime ideals, Maximal ideals

Lecture 12 - Maximal ideals

Lecture 13 - Hillbert’s Nullstellensatz

Lecture 14 - Hillbert’s Nullstellensatz (Continued...)

Lecture 15 - Application of Hillbert’s Nullstellensatz

Lecture 16 - Unique Factorization domian

Lecture 17 - Properties of Unique Factorization domain

Lecture 18 - Principal ideal domain

Lecture 19 - Properties of PID and ED

Lecture 20 - Properties of PID and ED (Continued...)

Lecture 21 - Prime elements of Z[i]

Lecture 22 - Prime elements of Z[i] (Continued...)

Lecture 23 - Application in Z[i]

Lecture 24 - Polynomial Rings over UFD

Lecture 25 - Gauss's Lemma

Lecture 26 - Polynomial Ring over UFD and Irreducibility Criterion

Lecture 27 - Irreducibility Criterion

Lecture 28 - Chinese Remainder Theorem

Lecture 29 - Nilradical and Jacobson radical

Lecture 30 - Examples and Problems

Lecture 31 - Definition of Modules and Examples

Lecture 32 - Definition of Modules and Examples (Continued...)

Lecture 33 - Submodules,direct sum and direct product of modules

Lecture 34 - Direct sum and direct product of modules, free modules

Lecture 35 - Finitely generated modules, free modules vs Vector spaces

Lecture 36 - Free modules vs Vector spaces

Lecture 37 - Vector spaces vs free modules and Examples

Lecture 38 - Quotient modules and module homomorphisms

Lecture 39 - Module homomorphism, Epimorphism theorem

Lecture 40 - Epimorphism theorem

Lecture 41 - Maximal submodules, minimal submodules

Lecture 42 - Freeness of submodules of a free module over a PID

Lecture 43 - Torsion modules, freeness of torsion-free modules over a PID

Lecture 44 - Rank of a module, p-submodules over a PID

Lecture 45 - Structure of a torsion module over a PID

Lecture 46 - Structure theorem, chain conditions

Lecture 47 - Artinian modules, Artinian rings

Lecture 48 - Noetherian modules, Noetherian rings

Lecture 49 - Ascending chain condition, Noetherian modules

Lecture 50 - Examples of Noetherian and Artinian modules and rings

Lecture 51 - Composition series, Modules of finite length

Lecture 52 - Jordan-Holderâ's theorem

Lecture 53 - Artinian rings

Lecture 54 - Noetherian rings

Lecture 55 - Hilbert basis theorem

Lecture 56 - Cohenâ's theorem on Noetherianness

Lecture 57 - Nakayama lemma

Lecture 58 - Nil and Jacobson radicals in Artinian rings

Lecture 59 - Structure theorem

Lecture 60 - Comparison between Artinian and Noetherian rings