# Modern Algebra

Learners enrolled: 2790
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 for the students at an undergraduate level. This course will serve as a useful tool to any learner who wishes to learn about modern algebra.

Summary
 Course Status : Completed Course Type : Core Duration : 8 weeks Start Date : 07 Aug 2019 End Date : 30 Sep 2019 Exam Date : 10 Nov 2019 IST Enrollment Ends : 10 Sep 2019 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. The general linear group

Week – III

9. The Group of symmetries
10.The subgroup generated by a subset
11.Cosets and Index of subgroup
12.Properties of group homomorphisms

Week – IV

13.Normal subgroups
14.Quotient groups
15.Class Equations
16.Direct product of a finite number of groups

Week – V

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

Week – VI

21.Rings from number systems
22.Ring of real quaternions & Rings of continuous functions
23.Rings of matrices
24.Polynomial rings

Week – VII

25.Subrings and ideals
26.Set of zero Divisors and group of units
27.Properties of Ring homomorphisms
28.Operations on ideals

Week – VIII

29.Integral domains and fields
30.Maximal ideals and Prime ideals
31.Euclidian Domains and Principal Ideal Domains
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.

### Instructor bio

Dr. T. Asir is working as an Assistant Professor and Head i/c in the Department of Mathematics-DDE, Madurai Kamaraj University, Tamil Nadu. 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 21 research articles in reputed/SCI Journals and a book with a citation count of 116 in Web of Science and 103 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 a MOOC course “Core and Pedagogy of Mathematics” which was launched at SWAYAM platform during 16th July 2018 to 24th August 2018.

### Course certificate

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