MA 4070: Elements of Groups and Rings

Update: The grades for MA 4070 have been submitted. If you wish to know your grade, then you may contact the academic section.
The grades are given based on your performance in the General Tests, Surprise Tests, Assignments, and the examinations.
All updates related to MA 4070 are announced here.

### Class timings:

Monday: 11:30 am to 1:00 pm
Wednesday: 08:30 am to 10:00 am
Saturday: 10:00 am to 11:30 am (Tutorial)

Date for the final exam: 22nd April 2015    Venue: 126    Time: 09:00 am to noon.

The grades will be decided by the marks obtained in the assignments, mid-term, surprise tests, the final examination, and also by the performance in the class. A google spread sheet, with all the marks obtained up to that point, was shared with you to assess yourself about your performance in the course1.

## Syllabus:

Binary operation and its properties, Definition of Groups, Examples and basic properties. Subgroups, Coset of a subgroup, Lagrange’s theorem. Cyclic groups. Normal subgroups, Quotient group. Homomorphisms, Isomorphism theorems. Permutation groups, Cayley’s theorems. Direct and semidirect product of groups. Group actions and Sylow theorems.

Definition of Rings, Examples and basic properties, Zero divisors, Integral domains, Fields, Characteristic of a ring, Quotient field of an integral domain. Subrings, Ideals, Quotient rings, Isomorphism theorems. Ring of polynomials. Prime, Irreducible elements and their properties, Unique Factorization Domains, Principal Ideal Domains, and Euclidean domains. Prime ideal, Maximal ideal, Prime avoidance theorem, Chinese remainder theorem.

## References:

• D. S. Dummit and R. M. Foote, Abstract Algebra, John Wiley & Sons Inc, 3rd Edition. 2004.
• I. N. Herstein, Topics in Algebra, Wiley India Pvt Ltd, 2nd Edition, 2006.
• Artin, M., Algebra, Prentice Hall of India, 1994.
• Jacobson, N., Basic Algebra I, Hindustan Publishing Corporation, 2nd Edition, 1991.

## Assignments:

The assignments have to be submitted right after the class on the respective due dates. The assignments can be submitted after the due date and time as well, but only 50% of the marks will be credited into your account, unless you have a valid reason for the late submission.

### Group Theory:

1. Assignment 1: Due on 28.01.2015.
2. Assignment 2: Due on 04.02.2015.
3. Problem Sheet 1: Discussed on 07.02.2015 (Tutorial)
4. Assignment 3: Due on 11.02.2015.
5. Assignment 4: Due on 18.02.2015.

### Ring Theory:

1. Assignment 5: Due on 23.03.2015.
2. Problem Sheet 2
3. Assignment 6: Due on 06.04.2015.
4. Assignment 7: Due on 17.04.2015.

Some of the assignments and tutorial sheets are prepared by Mr. Surjeet, a tutorial assistant for MA 4070 in Jan - May 2015.
The access for the spread sheet will be unavailable to students two days after the last class..