**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^{-1}R (resp. M and S^{-1}M)

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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