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**A first course in noncommutative rings.**
*(English)*
Zbl 0728.16001

Graduate Texts in Mathematics, 131. New York etc.: Springer-Verlag. xv, 397 p. DM 86.00 (1991).

Assuming a modest knowledge of abstract algebra taught in a standard first-year graduate course, this textbook gives a general introduction to the theory of rings. Consisting of eight chapters, it begins with the classical Wedderburn-Artin theory of semisimple rings (Chapter 1), and then deals with Jacobson’s general theory of the radicals for rings possibly without any chain conditions (Chapter 2). After a pleasant excursion into the rudiments of representation theory (Chapter 3), the author resumes his course with prime and primitive rings (Chapter 4), keeping with division rings (Chapter 5), ordered rings (Chapter 6), local and semilocal rings (Chapter 7), and finally ending up with perfect and semiperfect rings (Chapter 8). Such topics as the detailed exposition of the Mal’cev-Neumann construction of general Laurent series rings given here are not easily available in other standard textbooks on ring theory. The intended second volume will probably cover the theory of modules, rings of quotients, rings with polynomial identities, the structure theory of finite-dimensional central simple algebras, etc.

Reviewer: H.Nishimura (Ibaraki)

### MSC:

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16L30 | Noncommutative local and semilocal rings, perfect rings |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |