Much of Modern Algebra involves properties of sets together with operations. In the course on Modern Algebra, we have discussed about two major concepts namely Groups and Rings. Basically Group is a structure which involves a set with a single operation whereas Ring is a structure which involves a set with two operations. In the course on Modern Algebra we have covered the basic concepts of group theory and ring theory as extensively as possible. This course will serve as a useful tool to any learner who wishes to learn about Algebra, Linear Algebra, Topology and Algebraic Number Theory. Also, this course is essential for all leaners planning for advance degree in mathematics or planning to enter the teaching profession.
Week – I 1. Groups - Introduction2. Subgroups3. Elementary Properties of Abelian Groups4. Elementary Properties of Non Abelian Groups Week – II 5. The group of Integer Modulo n6. Complex Roots of Unity7. Cyclic Group8. General Linear Group Week – III 9. The Group of Symmetries10.Subgroups Generated by a Subset11.Cosets and Index of Subgroups12.Properties of Group Homomorphism and Isomorphism Week – IV 13.Normal Subgroups14.Quotient Groups15.Class Equation16.Direct Product of a Finite Number of Groups Week – V 17.Fundamental Theorem of Finite Abelian Groups18.Cayley's Theorem and Problems in Group Theory19.Rings - Introduction20.Class of Rings Week – VI 21.Rings from Number System22.The Ring of Real Quaternion's and Rings of Continuous Functions23.Ring of Matrices24.Polynomial Rings Week – VII 25.Subrings and Ideals26.Zero - Divisor and Group of Units27.Properties of Ring Homomorphism28.Operations on Ideals Week – VIII 29.Integral Domains and Fields30.Maximal Ideals and Prime Ideals31.Euclidean Domains and Principal Ideal Domain32.Unique Factorization Domains