# Rings and Modules

By Prof. Mousumi Mandal, Prof. Ramakrishna Nanduri   |   IIT Kharagpur, IIT Kharagpur
Learners enrolled: 526
This course is a self-contained elementary introduction to Rings and Modules. We will cover basic topics of Ring Theory and Module Theory which is a core course in Algebra. This course is compulsory to those who want to pursue research in Algebra. This is a 12 weeks course. In the first six weeks we focus on the Ring Theory part and the last six weeks we focus on the Module Theory part.

PRE - REQUISITES : Group Theory, Linear Algebra

INTENDED-AUDIENCE: All learneres interested in Mathematics
Summary
 Course Status : Completed Course Type : Elective Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Undergraduate/Postgraduate Start Date : 25 Jul 2022 End Date : 14 Oct 2022 Enrollment Ends : 08 Aug 2022 Exam Date : 30 Oct 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

WEEK 1 : Introduction to rings, examples, ideals, ring homomorphisms, quotients with examples
WEEK 2 : Isomorphism theorems, Prime and maximal ideals, basic properties of ideals, local rings
WEEK 4 : Integral domains, field of fractions, Euclidean Domains, Principal Ideal Domains
WEEK 5 : Unique factorization domains, Gauss lemma, Eisenstein’s irreducibility criterion
WEEK 6 : Algebraic variety, Weak form of Hilbert’s nullstellensatz
WEEK 7 : Introduction to Modules over arbitrary rings, examples, Free modules, submodules, quotient modules, module homomorphisms, direct sums, direct products
WEEK 8 : Generators of modules over commutative rings, rank of a free module, Simple modules
WEEK 9 : Modules over a PID
WEEK 10 : Modules with chain conditions, Noetherian modules, Noetherian rings
WEEK 11 : Artinian modules, Artinian rings, Hilbert Basis Theorem
WEEK 12 : Tensor product of modules, properties of tensor products

### Books and references

1.  M. Artin, Algebra, Pearson
2.  David S. Dummit and Richard M. Foote Abstract Algebra, John Wiley
3.  N. Jacobson, Basic Algebra, Volume 1, Dover
4. C. Musili, Introduction to Rings & Modules, Narosa.

### Prof. Mousumi Mandal

IIT Kharagpur
Prof. Mousumi Mandal is an Assistant Professor at the Department of Mathematics, IIT Kharagpur. Shedid her M.Sc and PhD from IIT Bombay. She works in the broad area of Commutative Algebra. After completing PhD she joined TIFR as a visiting Fellow. She was awarded the INdAM-COFUND Marie Curie Fellowship to pursue post doctoral work at the University of Genova, Italy. She was an INSA INSPIRE Fellow at IISc. Bangalore from 2013-2106. She joined IIT Kharagpur in 2016

### Prof. Ramakrishna Nanduri

IIT Kharagpur
Prof. Ramakrishna Nanduri did his M.Sc in Pure Mathematics from the University of Hyderabad. He did his Ph.D from IIT Madras. His broad research area is Commutative Algebra. He was awarded the INdAM-COFUND Marie Curie Fellowship to do post doctoral work at the University of Genova, Italy for a period of two years. Presently, he is an assistant professor at the Department of Mathematics, IIT Kharagpur since December 2014. He has various publications in reputed journals on various topics in Commutative Algebra and combinatorial aspects of Commutative Algebra.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 30 October 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
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. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

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 Kharagpur. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team