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Rings and Modules

Code: MA 621 | 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.


  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