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Indian Institute of Science Education and Research Bhopal
MTH 612: Non-commutative Algebra (4)
Pre-requisites: MTH 301, MTH 302, MTH 401
Matrix Rings and PLIDs, Tensor Products of Matrix Algebras, Ring constructions using Regular Representation
Basic notions for Noncommutative Rings, Structure of Hom(M,N), Semisimple Modules & Rings, the Wedderburn Structure Theorem, Simple Rings, Rings with Involution
The Jacobson Radical and its properties, Primitive Rings and Ideals, Hopkins-Levitzki Theorem, Nakayama’s Lemma , Radical of a Module, Local Rings, Chevalley-Jacobson Theorem, Kolchin’s Theorem, Clifford Algebras.
Prime and Semiprime rings, Rings of Fractions and Goldie’s Theorems, Rings with ACC (ideals), Tensor Algebras, Algebras over large Fields, Deformations and Quantum Algebras.
Hereditary Rings and their Modules, Division rings.
Central Simple Algebras, Cyclic Algebras, Symbol Algebras, Crossed Products, the Brauer Group, the functor Br, the Skolem-Noether Theorem, the centralizer Theorem, calculation of Brauer group of commutative rings.
Suggested Books:
- L. Rowen, Graduate algebra: noncommutative view, Graduate Studies in Mathematics, 91.
- B. Farb, R. Dennis, Noncommutative algebra, GTM, Springer-Verlag.
- T. Y. Lam, A first course in noncommutative rings, GTM, Springer.
- J. Golan and T. Head, Modules and the structure of rings: A primer, Pure and applied mathematics
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