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M.Phil Mathematics
Credit hours/ Marks:- (3 credits)
MP-654 Advanced Theory of Rings and Modules
Overview
Rings: Direct product of rings. Rings of polynomials. Fields of fractions. Factorization in integral domains. Sum and direct sum of ideals. Unique factorization domains. Principal ideal domains. Euclidean domains. Polynomial rings over UFD. Modules: R-homomorphism and quotient Modules. Direct sums and exact sequences. Free modules. Free modules over PIDs. Finitely generated modules over PIDs. Projective and injective modules. Completely reducible modules. Representation of linear mappings. Rank of a linear mapping. Irreducible polynomials and Eisenstein criterion. Adjunction of roots.
Credit hours/ Marks:- (3 credits)
1. Atiya, M.F. and Macdonal, I.G.: Introduction to Commutative Algebra (Addision Wesley Publishing Company, 1969). 2. Burton, D.E.: A First Course in Rings and Ideals (Addision Wesley Publishing Company, 1968). 3. Kaplansky, I.: Commutative Rings (Univ. of Chicago Press, 1974). 4. Vivek, S. and Vikas, B.: Algebra (Narosa Publishing House, India, 2003). 5. Bhattacharya, P. B., Jain, S. K. and Nagpaul, S. R.: Basic Abstract Algebra
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