Syllabus
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Course Code: Elective 6 MMATH21 -416 Course Name: 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 n (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 n (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. | |
