Syllabus
Course Code: MMATH21-416 Course Name: Elective VI) Non-Commutative Rings |
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MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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1 | Basic terminology and examples of non-commutative rings i.e. Hurwitz’s ring of integral quaternions, Free k-rings. Rings with generators and relations. Hilbert’s Twist, Differential polynomial rings, Group rings, Skew group rings, Triangular rings, D.C.C. and A.C.C. in triangular rings. Dedekind finite rings. Simple and semi-simple modules and rings. Spliting homomorphisms. Projective and Injective modules. Ideals of matrix ring M (R). Structure of semi simple rings. Wedderburn-Artin Theorem Schur’s Lemma. Minimal ideals. Indecomposable ideals. Inner derivation . -simple rings. Amitsur Theorem on non-inner derivations. |
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2 | Jacobson radical of a ring R.Annihilator ideal of an R-module M. Jacobson semi-simple rings. Nil and Nilpotent ideals. Hopkins-Levitzki Theorem. Jacobson radical of the matrix ring M (R). Amitsur Theorem on radicals. Nakayama’s Lemma. Von Neumann regular rings. E. Snapper’s Theorem. Amitsur Theorem on radicals of polynomial rings. | |
3 | Prime and semi-prime ideals. m-systems. Prime and semi-prime rings. Lower and upper nil radical of a ring R Amitsur theorem on nil radical of polynomial rings. Brauer’s Lemma. Levitzki theorem on nil radicals. Primitive and semi-primitive rings. Left and right primitive ideals of a ring R. Density Theorem. Structure theorem for left primitive rings. | |
4 | Sub-direct products of rings. Subdirectly reducible and irreducible rings. Birchoff’s Theorem. Reduced rings. G.Shin’s Theorem. Commutativity Theorems of Jacobson, Jacobson-Herstein and Herstein Kaplansky. Division rings. Wedderburn’s Little Theorem. Herstein’s Lemma. Jacobson and Frobenius Theorem. Cartan-Brauer-Hua Theorem. Herstein’s Theorem. ( 1.1 to 1.26, 2.1 to 2.9, 3.1 to 3.19, 4.1 to 4.27, 5.1 to 5.10, 10.1 to 10.30, 11.1 to 11.20, 12.1 to 12.11 and 13.1 to 13.26 of recommended book). |