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Noncommutative rings. (English) Zbl 0177.05801
The Carus Mathematical Monographs 15. Published by the Mathematical Association of America. Distributed by John Wiley and Sons, Inc., xii, 199 p. (1968).
This monograph gives an introduction to many aspects of classical ring theory. It is written in a clear style and introduces the reader to open questions as well as developed areas of the theory. The reader is assumed to have an introduction to algebraic structures such as groups, rings, fields, and vector spaces, and homomorphisms and quotient objects of such structures. The only background required beyond what would be obtained in almost any first course in abstract algebra is some knowledge of separable, normal, and inseparable field extensions. A bibliography is included at the end of each chapter for those desiring to read original sources.
The first topic discussed is the Jacobson radical, with applications to Artinian and semisimple Artinian rings. Maschke’s theorem is included in a sidetrip into group algebras. The density theorem for primitive rings is proved and applied to simple Artinian rings. The structure theory developed in the first part is then applied to obtain commutativity theorems such as Wedderburn’s result on finite division rings. Rings satisfying \(x^{n(x)}-x\) is central for all \(x\), are also shown to be commutative.
The next area covered is the Brauer group of simple algebras and crossed products. Wedderburn’s theorem is reproved in this context, and the classical theorems on real division algebras and ordered finite dimensional division algebras are obtained. Homological notions are introduced here to show the Brauer group is torsion. This section also includes complete reducibility of modules for a semisimple Artinian ring. (There is an unfortunate slip here of stating that one applies Zorn’s lemma to submodules expressible as direct sums of simples to get complete reducibility.)
Elementary representation theory of groups is the next topic covered, including a proof of Burnside’s theorem on the solvability of groups of order \(p^\alpha q^\beta\) and Frobenius’ theorem on groups containing a subgroup \(H\) such that \(H\cap x^{-1}Hx= \{1\}\) for all \(x\in H\).
The remaining areas explored are rings with polynomial identities and the Kurosh problem for them (using ultraproducts to get structure results for prime P.I. rings); Ore and Goldie theorems on quotient semisimple rings (using proofs of Procesi and Small); and the Golod-Shafarevich theorem with applications to the Kurosh and Burnside problems.

MSC:
16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras