# Modern Algebra

By Dr.T.Asir   |   Madurai Kamaraj University
Learners enrolled: 2521
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.
Summary
 Course Status : Ongoing Course Type : Core Duration : 8 weeks Start Date : 10 Jun 2020 End Date : 02 Aug 2020 Exam Date : Enrollment Ends : 15 Jul 2020 Category : Mathematics Credit Points : 4 Level : Undergraduate

### Course layout

Week – I

1. Groups - Introduction
2. Subgroups
3. Elementary Properties of Abelian Groups
4. Elementary Properties of Non Abelian Groups

Week – II

5. The group of Integer Modulo n
6. Complex Roots of Unity
7. Cyclic Group
8. General Linear Group

Week – III

9. The Group of Symmetries
10.Subgroups Generated by a Subset
11.Cosets and Index of Subgroups
12.Properties of Group Homomorphism and Isomorphism

Week – IV

13.Normal Subgroups
14.Quotient Groups
15.Class Equation
16.Direct Product of a Finite Number of Groups

Week – V

17.Fundamental Theorem of Finite Abelian Groups
18.Cayley's Theorem and Problems in Group Theory
19.Rings - Introduction
20.Class of Rings

Week – VI

21.Rings from Number System
22.The Ring of Real Quaternion's and Rings of Continuous Functions
23.Ring of Matrices
24.Polynomial Rings

Week – VII

25.Subrings and Ideals
26.Zero - Divisor and Group of Units
27.Properties of Ring Homomorphism
28.Operations on Ideals

Week – VIII

29.Integral Domains and Fields
30.Maximal Ideals and Prime Ideals
31.Euclidean Domains and Principal Ideal Domain
32.Unique Factorization Domains

### Books and references

1. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.

2. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

3. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson Prentice Hall, 2002.

4. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd Ed., John Wiley and Sons(Asia) Pvt. Ltd, Singapore, 2004.

5. Michael Artin, Abstract Algebra, 2nd Ed., Pearson Prentice Hall, 2011.

6. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995.

7. Robinson, Derek John Scott., An Introduction to Abstract Algebra, Hindustan Book Agency, 2010.

8. D.A.R.Wallace, Groups, Rings and Fields, Springer, 1998.