Modern Algebra

By Dr.T.Asir | Pondicherry University, Puducherry

Learners enrolled: 747

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 :	Completed
Course Type :	Core
Language for course content :	English
Duration :	8 weeks
Category :	Mathematics
Credit Points :	4
Level :	Undergraduate
Start Date :	31 Jul 2023
End Date :	24 Sep 2023
Enrollment Ends :	31 Aug 2023
Exam Date :

Page Visits

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.

Instructor bio

Dr.T.Asir

Pondicherry University, Puducherry

Dr. T. Asir is working as an Associate Professor in the Department of Mathematics, Pondicherry University. 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. His research interest is Algebra and Graph Theory. More specifically, graphs arising from rings and vice versa. He has published 36 research articles in International Journals (Journals like Discrete Mathematics, Discrete Applied Mathematics, Expositiones Mathematicae, Communications in Algebra, Soft Computing) and two books (including a book in Springer) with a citation count of 242 on Web of Science, 325 on Scopus and 521 on Google Scholar. He has completed three projects UGC-Startup grant, SERB-MATRICS project and Deanship of Scientific Research at King Abdulaziz University, Saudi Arabia. Currently, his research is supported by CSIR project. Also, he has received travel grants from DST(2012), UGC(2015), SERB(2019), CSIR(2020) and Gustave Eiffel University(2023) to deliver talks at International Conferences held at AlAin-UAE(2012), Thailand(2015), Palestine-Israel-Egypt(2019), Sharjah-UAE(2020) and France-Switzerland-Netherlands(2023). Further, 5 Ph.D.’s and 25 M.Phil.’s have been awarded under his guidance. Moreover, he has delivered more than 75 invited talks at various National/International conferences and Orientation/Refreshers programs. He has served as a referee of several international journals in the frame of pure and applied mathematics. Furthermore, he has developed online mathematical content through MOOC Courses “Core and Pedagogy of Mathematics”, “Modern Algebra” and “Graph Theory” in SWAYAM Platform (MHRD).

Course certificate

30 Marks will be allocated for Internal Assessment and 70 Marks will be allocated for end term proctored examinaton Securing 40% in both seperately is mandatory to pass the course and get Credit Certificate.

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