**ABOUT THE COURSE:**

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.

INTENDED AUDIENCE:

M.Sc. & Ph.D. Mathematics Students.

Linear Algebra, Basic group theory and Basic ring theory including ED, PID & UFD.

I am Associate Professor at the Department of Mathematics, IIT Madras. My area of research is Commutative Algebra.

**COURSE PLAN:**

MODULE 1 : Rings, ring homomorphism, ideals, quotients, zero divisors, nilpotents and units.

MODULE 2: Prime and maximal ideals, nilradical and Jacobsons radical

MODULE 3: Operations on ideals, extension and contraction.

MODULE 4: Modules and module homomorphisms, Submodules and quotient modules, Operations on submodules, Direct sum and product

MODULE 5: Finitely generated modules, Exact sequences, Tensor product of modules.

MODULE 6: Restriction and extension of scalars, Exactness properties of the tensor product.

MODULE 7: Localization

MODULE 8: Integral dependence, Going-up and Going-down theorems.

MODULE 9: Chain conditions, Noetherian rings

MODULE 10: Primary decomposition in Notherian rings.

MODULE 11: Artinian rings

**CERTIFICATION EXAM:**

- Course duration: 12 weeks
- Course start and end date: 23 January 2017 - 14 April 2017
- The exam is optional for a fee. Exams will be on
**23 April 2017.** - Time: Shift 1: 9am-12 noon; Shift 2: 2pm-5pm (Any one shift can be chosen to write the exam for a course.)
- Registration url: Announcements will be made when the registration form is open for registrations.
- The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published.

**CERTIFICATE:**

- Final score will be calculated as : 25% assignment score + 75% final exam score
- 25% assignment score is calculated as 25% of average of best 8 out of 12 assignments
- E-Certificate will be given to those who register and write the exam and score greater than or equal to 40% final score. Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT MADRAS. It will be e-verifiable at nptel.ac.in/noc.