This is an introductory course in Commutative Algebra where most basic tools on commutative rings and modules over commutative rings are developed. This course is essential for anyone who wants to do research in areas such as commutative algebra, algebraic geometry, algebraic number theory etc. INTENDED AUDIENCE : M.Sc. & Ph.D. Mathematics Students.PRE-REQUISITES : Linear Algebra, Basic group theory and Basic ring theory including ED, PID & UFD.
MODULE 1 : Rings, ring homomorphism, ideals, quotients, zero divisors, nilpotents and units.MODULE 2: Prime and maximal ideals, nilradical and Jacobsons radicalMODULE 3: Operations on ideals, extension and contraction.MODULE 4: Modules and module homomorphisms, Submodules and quotient modules, Operations on submodules, Direct sum and productMODULE 5: Finitely generated modules, Exact sequences, Tensor product of modules.MODULE 6: Restriction and extension of scalars, Exactness properties of the tensor product.MODULE 7: LocalizationMODULE 8: Integral dependence, Going-up and Going-down theorems.MODULE 9: Chain conditions, Noetherian ringsMODULE 10: Primary decomposition in Notherian rings.MODULE 11: Artinian rings
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Aoi completed this course.
Covers standard topics of introductory commutative algebra. Basic knowledge of abstract algebra (ring, field, ideal) is needed to understand the topics. What I find lacking is the (geometric) motivations behind many of the technique and theorems. Overall it is a very organized and easy to follow introductory course.