Course Archives Theoretical Statistics and Mathematics Unit | |||||
Course: Algebra I Level: Postgraduate Time: Currently not offered |
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Syllabus Past Exams Syllabus: i) Commutative rings with unity: examples, ring homomorphisms, ideals, quotients, isomorphism theorems with applications to non-trivial examples. Prime and maximal ideals, Zorns Lemma and existence of maximal ideals. Product of rings, ideals in a finite product, Chinese Remainder Theorem. Prime and maximal ideals in a quotient ring and a finite product. Field of fractions of an integral domain. Irreducible and prime elements; PID and UFD. ii) Polynomial Ring: universal property; division algorithm; roots of polynomials. Gausss Theorem (R UFD implies R[X] UFD); irreducibility criteria. Symmetric polynomials: Newtons Theorem. Power Series. iii) Modules over commutative rings: examples. Basic concepts: submodules, quotients modules, homomorphisms, isomorphism theorems, generators, annihilator, torsion, direct product and sum, direct summand, free modules, finitely generated modules, exact and split exact sequences. iv) Noetherian rings and modules, algebras, finitely generated algebras, Hilbert Basis Theorem. Tensor product of modules: definition, basic properties and elementary computations. Time permitting, introduction to projective modules. Suggested Texts : (a) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint 2003). (b) N. Jacobson, Basic Algebra Vol. I, W.H. Freeman and Co (1985). (c) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004). (d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986). (e) N.S. Gopalakrishnan, Commutative Algebra (Chapter 1), Oxonian Press (1984). Top of the page Past Exams | |||||
Midterm
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