Syllabus
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Course Code: MMATH21-305 Course Name: Elective - I ) Commutative Algebra |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Free module, submodules, cyclic modules, homomorphism of R-modules, rank of Module. exact sequence, projective modules, Shanuel’s lemma, tensor products, finitely generated R-algebra, flat modules. | |
| 2 | Ideals, maximal ideals, prime ideals, nilpotent elements, nil radical, Jacobson radical of R, comaximal, Chinese remainder theorem, extension and contraction of ideal, local rings, Nakayama lemma, localisation and quotients, localisation of localisation, applications, patching up of localisations. | |
| 3 | Noetherian modules, Hilbert’s basis theorem, primary ideal, primary decomposition. first and second uniqueness theorem, Artinian modules, structure of Artinian rings, composition series of R-module, Jordan Holder theorem, length of a module. | |
| 4 | Integral elements, integral closure, integral extensions, lying above, going up theorem, integrally closed domains, going-down theorem, finiteness of integral closure, Noether’s normalisation theorem, weak nullstellensatz, Hilbert’s nullstellensatz. (Chapter 1, 2, 3 & 4 of the recommended book) |
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