## A First Course in Noncommutative RingsThis book, an outgrowth of the author¿s lectures at the University of California at Berkeley, is intended as a textbook for a one-semester course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semisimple rings, Jacobson¿s theory of the radical, representation theory of groups and algebras, prime and semiprime rings, local and semilocal rings, perfect and semiperfect rings, etc. By aiming the level of writing at the novice rather than the connoisseur and by stressing the role of examples and motivation, the author has produced a text that is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition. |

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### Contents

2 Semisimplicity | 48 |

3 Structure of Semisimple Rings | 116 |

8 Representations of Groups | 117 |

9 Linear Groups | 141 |

CHAPTER 2 | 147 |

Jacobson Radical Theory OIIOCM J OOOOIIUJO | 154 |

12 Subdirect Products and Commutativity Theorems | 191 |

5 Jacobson Radical Under Change of Rings | 214 |

CHAPTER 6 | 261 |

6 Group Rings and the JSemisimplicity Problem | 276 |

CHAPTER 7 | 279 |

20 Semilocal Rings | 296 |

7 Modules over FiniteDimensional Algebras | 306 |

22 Central Idempotents and Block Decompositions | 326 |

Perfect and Semiperfect Rings | 335 |

25 Principal Indecomposables and Basic Rings | 359 |

l4 Some Classical Constructions | 216 |

15 Tensor Products and Maximal Subﬁelds | 238 |

Name Index | 373 |

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acts algebra apply artinian assume called centrally characteristic clearly commutative completely conclusion conjugate consider construction contains Corollary cyclic decomposition deﬁned deﬁnition denotes dimp direct division ring domain easily easy element equivalent example Exercise exists extension fact factor ﬁeld ﬁnite ﬁrst formally give given group G hence holds idempotents implies indecomposable inﬁnite instance integer irreducible isomorphism Jacobson left ideal left primitive left R-module Lemma linear matrix maximal minimal module multiplication nilpotent noetherian noncommutative nonzero Note notion ordered particular polynomial prime projective Proof properties Proposition prove radical reduced representation resp result right R-module root semilocal ring semiprime semisimple simple space splitting structure subﬁeld subgroup submodules subring Theorem theory unique unit write zero