Rings and Modules

Code: MA621 | L-T-P-C: 3-0-0-6

 Prerequisites: MA 521 (Modern Algebra) or equivalent

Brief review of rings and ideals, nilradical and Jacobson radicals, extension and contraction; basic theory of modules: submodules and quotient modules, module homomorphisms, annihilators, torsion submodules, irreducible modules, Schur's lemma, direct sum and product of modules, free modules, localization, Nakayama's lemma; Exact sequences, short and split exact sequences, projective modules, injective modules, Baer's criterion for injective modules; tensor product of modules, universal property of tensor product, exactness property of tensor products, flat modules; chain conditions on rings, Noetherian rings, Hilbert basis theorem; Artinian rings, discrete valuation rings, Dedekind domains, fractional ideals, ideal class groups.

Texts/References:

1. M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison Wesley, 1969.
2. D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley & Sons, Inc., II Edition, 1999.
3. S. Lang, Algebra, III edition, Springer, 2004.
4. P A. Grillet, Abstract Algebra, II edition, Springer 2006
5. T .W. Hungerford, Algebra, Springer, 1996