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.

Week

Topics

1

Rings, ring homomorphism, ideals, quotients, zero divisors, nilpotents and units.

2

Prime and maximal ideals, nilradical and Jacobsons radical

3

Operations on ideals, extension and contraction.

4

Modules and module homomorphisms, Submodules and quotient modules, Operations on submodules, Direct sum and product

5

Finitely generated modules, Exact sequences, Tensor product of modules.

6

Restriction and extension of scalars, Exactness properties of the tensor product.

7

Localization

8

Integral dependence, Going-up and Going-down theorems.

9

Chain conditions, Noetherian rings

10

Primary decomposition in Notherian rings.

11

Artinian rings

12

Linear Algebra, Basic group theory and Basic ring theory including ED, PID and UFD. Introduction to Commutative Algebra by M. F. Atiyah and I. G. Macdonald

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