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Indian Institute of Science Education and Research Bhopal
MTH 415: Commutative Algebra (4)
Pre-requisites: MTH 401 and its pre-requisites
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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