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Algebra

Course Code: MA 501; MM 801  |  Credits: 5  |  Prerequisite: Algebra-I, Algebra-II, Algebra-III

Syllabus Details

  • Unit I:(Modules) Review of modules, free modules, simple and semi-simple modules, isotypical components, the Artin-Wedderburn theorem; projective, injective and flat modules, finite presentation of a module, and the result that a finitely presented flat module is projective.
  • Unit II:(Artinian and Noetherian Rings) Artinian and Noetherian rings and modules; Jacobson and nil radicals and the structure of Artinian rings; modules over Noetherian rings, support of a module and associated prime ideals, and the result that every associated prime ideal lies in the support of a finitely generated module over a Noetherian ring.
  • Unit III:(Dimension and Dedekind Domains) Spectrum of a commutative ring, Krull dimension, height and coheight of a prime ideal, the relation between the Krull dimension of a ring and that of its polynomial ring; discrete valuation rings, fractionary ideals, Dedekind domains and their equivalent definitions.
  • Unit IV:(Graded Rings and Dimension Theory) Graded rings and graded modules, the Hilbert-Samuel polynomial, ideal of definition of a Noetherian local ring, I-adic filtrations and the associated graded ring, independence of the degree of the Hilbert polynomial from the ideal of definition, Chevalley dimension and the dimension theorem.

Course Learning Objective

  1. Understand the basic theory of commutative algebra.
  2. Understand and solve basic problems in commutative algebra.
  3. Apply these concepts to related areas such as algebraic geometry and algebraic number theory.
  1. Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Addison-Wesley, 1969
  2. Dummit, David S. and Foote, Richard M. Abstract Algebra. 3rd ed. Wiley, 2003
  3. Lang, Serge. Algebra. Revised 3rd ed. Graduate Texts in Mathematics 211. Springer, 2002
  4. Jacobson, Nathan. Basic Algebra I. 2nd ed. W. H. Freeman, 1985
  5. Jacobson, Nathan. Basic Algebra II. 2nd ed. W. H. Freeman, 1989
  6. Artin, Michael. Algebra. 2nd ed. Pearson, 2011
  7. Zariski, Oscar and Samuel, Pierre. Commutative Algebra I. Springer, 1975
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