Title Groups and Rings (Code 09010502)
|S.N||Unit||Content||Domain||Hours as per UGC|
|1||one||1. Definition of a group with example and simple properties of groups
2. Subgroups and subgroup criteria
3. Generation of groups, cyclic groups
4. Cosets, Left and right cosets
5. Index of a sub-group, Coset decomposition
6. Lagrange’s theorem and its consequences
7. Normal subgroups, Quotient groups
2. Automorphisms of cyclic groups Groups
|3||Three||1. Introduction to rings, subrings, integral domains and fields,
2. Characteristics of a ring. Ring homomorphisms, ideals (principle, prime and Maximal) and Quotient rings, Field of quotients of an integral domain.
3. Euclidean rings, Polynomial rings, Polynomials over the rational field, , Polynomial rings over Unique factorisation domain
4. R unique factorization domain implies so is R[X1,X2,…Xn] commutative rings,
Nice to know
- N. Herstein : Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975,
- B. Bhattacharya, S.K. Jain and S.R. Nagpal : Basic Abstract Algebra (2nd edition).
- Vivek Sahai and Vikas Bist : Algebra, NKarosa Publishing House.