Lecture 43 of 60Week 9
Lecture-43 : Torsion modules, freeness of torsion-free modules over a PID
Prof. Mousumi Mandal and Prof. Ramakrishna Nanduri · IIT Kharagpur
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- Rings and Modules, IIT Kharagpur (at NPTEL)
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All the lectures
- Lecture-1 : Introduction to Rings
- Lecture-2 : Rings, Subrings
- Lecture-3 : Ring Homomorphism, Ideals
- Lecture-4 : Properties of Ideals
- Lecture-5 : Properties of Ideals(contd.)
- 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 (contd.)
- 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(contd.)
- Lecture-21 : Prime elements of Z[i]
- Lecture-22 : Prime elements of Z[i] (contd.)
- 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(contd.)
- 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 & 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-Holder’s theorem
- Lecture-53 : Artinian rings
- Lecture-54 : Noetherian rings
- Lecture-55 : Hilbert basis theorem
- Lecture-56 : Cohen’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 & Noetherian rings