The theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This volume is a continuation and an in-depth study, stressing the non-commutative nature of the first two volumes of Algebras, Rings and Modules by M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. It is largely independent of the other volumes. The relevant constructions and results from earlier volumes have been presented in this volume.

Rings, which play a fundamental role in analysis, geometry, and topology, constitute perhaps the most ubiquitous algebraic structures across mathematics. Starting with fields (the most familiar rings), considering polynomials leads to commutative rings and considering matrices leads to non-commutative algebra. The present volume, though lacking a number, joins a series now spread over three publishers. Like volumes 1 and 2, it surveys aspects of non-commutative rings (volume 3 went in another direction) but stands self-contained, despite the occasional reference to previous volumes. Progress in ring theory depends on conditions designed to isolate special classes of rings admitting satisfying structure theorems. Each chapter surveys such conditions and their consequences: hereditary rings, valuation domains, nonsingular rings, Goldie rings, FDI-rings, exchange rings, Rickart rings, serial nonsingular rings, and many more. Commutative ring concepts tend to have diverse, but limited, generalizations in the non-commutative context, and the early chapters particularly explore such themes. The authors supply complete details, even for simple arguments that other authors might package into exercises (of which this book has none), making the book an excellent reference. Readers will find all developments digested into small satisfying steps, with major results seeming to drop out effortlessly. --D. V. Feldman, University of New Hampshire, Appeared in February 2017 issue of CHOICE