Lecture-1:Introduction to Rings

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Lecture-2: Rings, Subrings.

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Lecture-3:Ring Homomorphism, Ideals.

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Lecture-4:Properties of Ideals.

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Lecture-5:Properties of Ideals (contd.)

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Lecture-06: Quotient Ring, Isomorphism Theorem

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Lecture-07:Isomorphism Theorem, Homomorphism Theorem

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Lecture-08: Homomorphism Theorem

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Lecture-09: Integral Domain, Quotient Ring

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Lecture-10: Quotient Ring

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Lecture-11: Prime ideals, Maximal ideals

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Lecture-12: Maximal ideals

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Lecture-13: Hillbert’s Nullstellensatz

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Lecture-14: Hillbert’s Nullstellensatz (contd.)

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Lecture-15 : Application of Hillbert’s Nullstellensatz

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Lecture-16: Unique Factorization domian

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Lecture-17: Properties of Unique Factorization domain

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Lecture-18: Principal ideal domain.

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Lecture-19: Properties of PID and ED

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Lecture-20: Properties of PID and ED(contd.)

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Lecture-21: Prime elements of Z[i]

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

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Lecture-23: Application in Z[i]

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Lecture-24 : Polynomial Rings over UFD

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Lecture-25 : Gauss's Lemma

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Lecture-26: Polynomial Ring over UFD and Irreducibility Criterion

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Lecture-27: Irreducibility Criterion

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Lecture-28: Chinese Remainder Theorem

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Lecture-29: Nilradical and Jacobson radical

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Lecture-30: Examples and Problems

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Lecture-31 : Definition of Modules and Examples

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Lecture-32: Definition of Modules and Examples(contd.)

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

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

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

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Lecture-36: free modules vs Vector spaces

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Lecture-37: Vector spaces vs free modules & Examples.

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Lecture -38: Quotient modules and module homomorphisms.

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Lecture-39: Module homomorphism , Epimorphism theorem

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Lecture-40: Epimorphism theorem.

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Lecture-41: Maximal submodules, minimal submodules

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

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

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

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Lecture-45: Structure of a torsion module over a PID

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Lecture-46: Structure theorem, chain conditions

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Lecture-47: Artinian modules, Artinian rings

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Lecture-48: Noetherian modules, Noetherian rings

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Lecture-49: Ascending chain condition, Noetherian modules

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Lecture-50: Examples of Noetherian and Artinian modules and rings

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Lecture-51: Composition series, Modules of finite length

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Lecture-52: Jordan-Holder’s theorem

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Lecture-53: Artinian rings

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Lecture-54: Noetherian rings

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Lecture-55: Hilbert basis theorem

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Lecture-56 : Cohen’s theorem on Noetherianness

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Lecture-57: Nakayama lemma

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Lecture-58: Nil and Jacobson radicals in Artinian rings

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Lecture-59: Structure theorem

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Lecture-60: Comparison between Artinian & Noetherian rings

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Live Session 18-10-2023

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Live Session 27-09-2023

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Live Session 30-08-2023

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