1800 102 5661

### Title Groups and Rings (Code 09010502)

 S.N Unit Content of the topics Learning objects Teaching Guidelines Methodology Time (Hrs) 1 one Definition of a group with example and simple properties of groups Subgroups and subgroup criteria Generation of groups, cyclic groups Cosets, Left and right cosets Index of a sub-group, Coset decomposition Lagrange’s theorem and its consequences Normal subgroups, Quotient groups Students will be able to 1. Geometrical meaning of DE. 2. First order higher degree equations solvable for x,y,p.Lagrange’s equations 3. Clairaut’s equations Lecture should be effective so that student can grasp the topics easily. Lecture.  Seminar. Discussion/Interaction with Students. Assignment.  Discussion on Assignment. Evaluation of Assignment. 15 2 Two Homoomorphisms Automorphisms of cyclic groups Groups Homoomorphisms Students will be able to Orthogonal trajectories in cartesian coordinates and polar coordinates 2. Linear DE with constant coefficients. Homogeneous linear ODE. Transformation of the equation by changing the dependent variable/independent variable. Method of variations of parameters Lecture should be effective so that student can grasp the topics easily. Lecture.  Seminar. Discussion/Interaction with Students. Assignment. Discussion on Assignment. Evaluation of Assignment 15 3 Three Introduction to rings, subrings, integral domains and fields, Characteristics of a ring. Ring homomorphisms, ideals (principle, prime and Maximal) and Quotient rings, Field of quotients of an integral domain. Euclidean rings, Polynomial rings, Polynomials over the rational field, , Polynomial rings over Unique factorisation domain R unique factorization domain implies so is R[X1,X2,…Xn] commutative rings, To make students familiar with Ordinary simultaneous DE, Solution of simultaneous DE involving operators x (d/dx) or t (d/dt) etc. General method of solving Pdx+Qdy+Rdz=0 by taking one variable constant. Lecture should be effective sothat student can grasp the topics easily. Lecture. Seminar. Discussion/Interaction with Students. Assignment. Discussion on Assignment. Evaluation of Assignment 15