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Syllabus Ring Theory

Syllabus :

  1. Definition and properties of a Ring, Subring. [5 Lectures]
  2. Integral Domains : Zero devisiors,  Cancellation Law, Field, Characteristics of Ring. [5 Lectures]
  3. Ideals and Factor Rings : Existence of Factor Ring, Prime Ideals, Maximal Ideals.  [6 Lectures]
  4. Homomorphism of Rings : Properties of Ring Homomorphism, Kernel, First isomorphism Theorem for Ring, Prime Fields. The field of Quotients. [8 Lectures]
  5. Polynomial Ring : Definition. The division Algorithm, Principle Ideal Domain. [6 Lectures]
  6. Factorization of Polynomial : Reducibility and Irreducibility Tests, Eisenstein criterion. Ideals in F[x]: Unique Factorization in Z[x]. [8 Lectures]
  7. Divisibility in Integral Domain: Associates, Irreducible and Primes, Unique Factorization Domains, Ascending chain Condition for PID, PID implies UFD, Euclidean Domains. ED Implies PID, D is UFD implies D[x] is UFD. [10 Lectures]

Text Book:

  • Joseph, A. Gallian, Contemporary Abstract Algebra,(4th Edition), Narosa Publishing House.
  • Chapter Numbers : 12,13,14,15,16,17 and 18.

Reference Books:

  • J.B. Fraleigh, First course in Abstract Algebra (4rd Edition). Narosa Publishing House.
  • I.N. Herstein. Abstract Algebra, (3rd Edition), Prentice Hall of India, 1996.
  • N.S. Gopalkrishnan, University of Algebra, Wiley Eastern 1986.
  • C. Musili, Rings and Modules, Narosa Publishing House, 1992.