Noncommutative Algebra
نویسنده
چکیده
1. Basics 2 1.1. Commutants 2 1.2. Opposite Rings 3 1.3. Units 3 1.4. Ideals 4 1.5. Modules 4 1.6. Division rings 5 1.7. Endomorphism rings 7 1.8. Monoid Rings 8 1.9. Noetherian and Artinian Rings 11 1.10. Simple rings 15 1.11. Prime ideals and prime rings 17 1.12. Notes 18 2. Wedderburn-Artin Theory 18 2.1. Semisimple modules and rings 18 2.2. Wedderburn-Artin I: Semisimplicity of Mn(D) 20 2.3. Wedderburn-Artin II: Isotypic Decomposition 21 2.4. Wedderburn-Artin III: When Simple Implies Semisimple 25 2.5. Maschke’s Theorem 26 2.6. Notes 30 3. Radicals 30 3.1. Radical of a module 30 3.2. Nakayama’s Lemma 32 3.3. Nilpotents and the radical 34 3.4. The Brown-McCoy radical 35 3.5. Theorems of Wedderburn and Kolchin 35 3.6. Akizuki-Levitzki-Hopkins 36 3.7. Functoriality of the Jacobson radical 37 3.8. Notes 40 4. Central Simple Algebras I: The Brauer Group 41 4.1. First properties of CSAs 41 4.2. The Brauer group 43 4.3. The Skolem-Noether Theorem 45 4.4. The Double Centralizer Theorem 46 4.5. Notes 47
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