Description and Objective

This is a course in advanced abstract algebra, which builds on the concepts learnt in Algebra I. The objectives of the course are to introduce students to the basic ideas and methods of modern algebra and enable them to understand the idea of a ring and an integral domain, and be aware of examples of these structures in mathematics; appreciate and be able to prove the basic results of ring theory; appreciate the significance of unique factorization in rings and integral domains

Prerequisite

 Algebra-I

Learning Outcomes

Upon successful completion of this module students will be able to

• Demonstrate knowledge of the syllabus material;
• Write precise and accurate mathematical definitions of objects in ring theory;
• Use mathematical definitions to identify and construct examples and to distinguish examples from non-examples;
• Validate and critically assess a mathematical proof;
• Use a combination of theoretical knowledge and independent mathematical thinking to investigate questions in ring theory and to construct proofs;
• Write about ring theory in a coherent, grammatically correct and technically accurate manner.

Contents

1. Rings: Definition, examples. Quadratic integer rings,
2. Examples of non-commutative rings,
3. The Hamilton quaternions. Polynomial rings. Matrix rings. Units, zero-divisors,
4. nilpotents, idempotents. Subrings, Ideals,
5. Maximal and prime Ideals. Left, right and two-sided ideals; Operations with ideals,
6. The ideal generated by a set. Quotient rings. Ring homomorphism,
7. The isomorphism theorems, applications. Finitely generated ideals. Rings of fractions,
8. Integral Domain: The Chinese remainder theorem.
9. Divisibility in integral domains,
10. greatest common divisor, least common multiple.
11. Euclidean domains,
12. The Euclidean algorithm,
13. Principal ideal domains,
14. Prime and irreducible elements in an integral domain.
15. Gauss lemma,
16. irreducibility criteria for polynomials

 Recommended Books

1. Gallian, J.A., Contemporary Abstract Algebra, 9th ed. (Brooks/Cole, (2017).
2. Malik D. S., Mordeson J. N. and Sen M. K., Fundamentals of Abstract Algebra, (WCB/McGraw-Hill, 1997).

 Suggested Books

1. Roman, S., Fundamentals of Group Theory, 1st edition, (Birkhäuser Basel, (2012)).
2. Rose, H. E. A Course on Finite Groups, 1st edition, (Springer-Verlag London, (2009)).
3. Rotman, J.J., An Introduction to the theory of groups, 4th edition (Springer; (1999)).
4. Rose, J., A Course on Group Theory, revised edition, (Dover Publications, (2012)).
5. Fraleigh, J.B., A First Course in Abstract Algebra, 7th edition (Addison-Wesley Publishing Company, (2003)).

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

The projects assigned in this course follow as assignments the exercises related to the topics from the suggested book. 

Assessment Criteria

       Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

       Mid-Term Exam:   30

       Final-Term Exam: 50

Key Dates and Time of Class Meeting

Monday                                                   12:30 PM-02:00 PM 

Wednesday                                              09:30 PM-11:00 PM

Thursday                                                  12:30 PM-02:00 PM 

Friday                                                        11:00 PM-12:30 PM