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Indian Institute of Science Education and Research Bhopal

MTH 618: Commutative Algebra (4)

Pre-requisites (Desirable): MTH 401: Fields and Galois Theory

Learning Objectives:

The aim of this course is to introduce commutative algebra. This theory has developed not just as a standalone area of algebra, but also as a tool to study other important branches of Mathematics including Algebraic Geometry and Algebraic Number Theory.

Course Contents:

• Quotient Rings, Prime and Maximal ideals, units, Nilradical, Jacobson Radical, Operations on ideals, Extensions and contractions
• Tensor product of Algebras (only existence theorem), Rings and Modules of fractions, Local properties, Structure passing between R and S^-1R (resp. M and S^-1M)
• Primary decompositions, Uniqueness theorems, Chain conditions, Noetherian and Artinian Rings, Lasker-Noether theorem, Hilbert basis theorem, Nakayama's lemma, Krull intersection theorem
• Integral dependence, Going up theorem, Integrally closed integral domains, Going down theorem
• Valuation rings, Discrete valuation rings, Dedekind domains, Fractional ideals
• Valuations, Completions, Extensions of absolute values, residue field, Local fields, Ostrowski's theorem
• Hilbert's Nullstellensatz

Suggested Books:

• Introduction to Commutative Algebra, Atiyah, M and Macdonald, I.G., Levant Books, Kolkata
• Graduate Algebra: Commutative View, Rowen, L.H., Graduate Studies in Mathematics, AMS
• Commutative Algebra with a view towards Algebraic Geometry, Eisenbud, D., Springer

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