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.

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

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.

Dr. T. Asir is working as an Assistant Professor and Head i/c in the Department of Mathematics-DDE, Madurai Kamaraj University, Tamil Nadu. Also he is the Director (FAC) for MKU Evening College, Dindigul. He has received a gold medal in M.Sc. and secured second rank in M.Phil. examinations. His Ph.D. work was supported by the grants “Major Project Fellowship” by UGC and “INSPIRE Fellowship” by DST. He did Post Doctorate under the UGC-Kothari Postdoctoral Fellowship. He has published 24 research articles in reputed/SCI Journals and a book with the citation count of 145 in Web of Science and 151 in Scopus. He has completed a UGC-StartUp grant during 2015-17. Currently his research work is supported by SERB through MATRICS grant. Also 2 P.hd’s and 21 M.Phil’s have been awarded under his guidance. Further he has developed MOOC courses “Core and Pedagogy of Mathematics”(16th July 2018 to 24th August 2018), “Modern Algebra”(7th Aug. 2019 to 30th Sep 2019) and “Graph Theory”(22nd January 2020 to 11th April 2020) at SWAYAM platform.

30% for in course Assessment & 70% of end term Proctored Exam

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