Commutative Algebra

Syllabus Details

• Unit I:(Review of Rings and Modules) Rings (commutative rings with unity), ring homomorphism, ideals, quotient rings, prime ideals, maximal ideals, nil radical, Jacobson radical, extension and contraction of ideals, modules, module homomorphism, submodules, quotient modules, direct sum and direct product of modules, finitely generated modules, Nakayama lemma, exact sequence.
• Unit II:(Tensor Product) Tensor product of modules-construction and universal property, examples, associativity of tensor product, extension and restriction of scalars, compatibility of exact sequence and tensor product, flat modules and their characterization, faithfully flat modules, tensor product of algebras.
• Unit III:(Localization and Primary Decomposition) Multiplicatively closed sets, ring of fractions and module of fractions, examples, localization at a prime ideal, fraction operation on ideals, prime ideals in ring of fractions, compatibility of fraction operation with exact sequence, tensor product, local properties, primary ideals, primary decomposition, uniqueness theorems of primary decomposition.
• Unit IV:(Integral Extension) Integral element, ring of integers, integral extension, going-up theorem, integrally closed domain, going-down theorem, valuation rings, Hilbert’s Nullstellensatz, Noether normalization lemma.
• Unit V:(Noetherian and Artinian Rings/Modules) Ascending and descending chain conditions, Noetherian and Artinian rings/modules and their basic properties, ring of fractions of Noetherian rings, Hilbert basis theorem, primary decomposition in Noetherian rings.
• Unit VI:(Discrete Valuation, Dedekind Domain and Dimension Theory) Discrete valuation and discrete valuation rings, Noetherian integrally closed local domains, Dedekind domains, fractional ideals, factorization of ideals in Dedekind domain, Krull dimension of a ring, dimension of a module, Chevalley dimension, dimension theorem for finitely generated module over a Noetherian local ring, Krull’s principal ideal theorem.

Course Learning Objective

1. Employ basic operations like tensor product, localization to find out characteristics of given module.
2. Understand concept related to primary decomposition.
3. Visualize the extension of prime ideals in integral extension, particularly in the context of Dedekind domains.
4. Understand underlying concepts behind Hilbert’s Nullstellensatz.
5. Gain a general idea, how the concepts of Commutative Algebra appear in Algebraic Number Theory and Algebraic Geometry.

Recommended Books

1. Atiyah, M. F., and Macdonald, I. G. Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969
2. Zariski, O. and Samuel, P. Commutative Algebra, Vol. I. GTM No. 28, Springer-Verlag, New York, 1958
3. Matsumura, Hideyuki. Commutative Algebra. 2nd ed. Mathematics Lecture Note Series, 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980
4. Gopalakrishnan, N. S. Commutative Algebra. Oxonian Press, 1984
5. Reid, M. Undergraduate Commutative Algebra. Cambridge University Press, 1995