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Swayam Central

Modern Algebra

By Dr. T. Asir   |   Madurai Kamaraj University, Madurai, Tamil Nadu.
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.


Learners enrolled: 2768

SUMMARY

Course Status : Ongoing
Course Type : Core
Duration : 8 weeks
Start Date : 07 Aug 2019
End Date : 30 Sep 2019
Exam Date : 10 Nov 2019
Enrollment Ends : 10 Sep 2019
Category :
  • Mathematics
  • 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

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