Mathematics

**MTH ****302: Rings and Modules**** ****(4)**

**Pre-requisites:** MTH 301

*Learning Objectives*:

This is an introductory course on Group theory. We will begin by studying the basic concepts of subgroups, homomorphisms and quotient groups with many examples. We then study group actions, and prove the Class equation and the Sylow theorems. They are in turn used to prove the structure theorem for finite abelian groups and to discuss the classification of groups of small order. We then turn to solvability, prove the Jordan-Holder theorem, and discuss nilpotent groups (if time permits).

*Course Contents*:

- Definition of rings, Homomorphisms, basic examples (Polynomial ring, Matrix ring, Group ring), Integral domain, field, Field of fractions of an integral domain
- Ideals, Prime and Maximal ideals, Quotient Rings, Isomorphism theorems, Chinese Remainder theorem, Applications
- Principal ideal domains, Irreducible elements, Unique factorization domains, Euclidean domains, examples
- Polynomial rings, ideals in polynomial rings, Polynomial rings over fields, Gauss’ Lemma, Polynomial rings over UFDs, Irreducibility criteria
- Definition of modules, submodules, The group of homomorphisms, Quotient modules, Isomorphism theorems, Direct sums, Generating set, free modules, Simple modules, vector spaces
- Free modules over a PID, Finitely generated modules over PIDs
- Applications to finitely generated abelian groups and Rational and Jordan canonical forms
- (if time permits) Tensor product of modules, Exact sequences of modules, Hom functor, Projective modules, Injective modules, Baer’s criterion

*Suggested Books*:

- D.S. Dummit, R.M. Foote, Abstract Algebra, 2
^{nd}Edition, Wiley - G. Birkhoff, S. McLane, Algebra (3
^{rd}Edition), AMS - S. Lang, Algebra (3
^{rd}Edition), Pears - C. Musili, Rings and Modules (2
^{nd}Edition), Narosa - M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra (1
^{st}Indian Edition), Levant Books - N. Jacobson, Basic Algebra (Vols - I & II), Hindustan Book Agency

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