This book is intended for the general reader of mathematics, and the authors have eliminated unnecessary mathematical machinery whenever possible, or postponed using it as long as possible. This was done, however, only after the authors saw that it could be done without any loss of clarity, or generality, in the statements of theorems.

The book is as elementary and self-contained as practicable; and the little background required in homological and categorical algebra is listed in two short appendices. Otherwise, full definitions are given, and short, elementary full proofs are supplied for such tool theorems as the Morita theorem 1.12 (p. 15), the correspondence theorem 1.13 (p. 17), the Wedderburn–Artin theorem 2.23 (p. 32), the Goldie–Lesieur–Croisot theorem 4.4 (p. 65), and many others.

The authors are indebted to Professor Hyman Bass, one of the general editors of this series, for suggesting that such a study of simple Noetherian rings would be of interest. We are immensely grateful to Professor Bass and to the Cambridge University Press for making it possible for this study to appear in the Cambridge Tracts in Mathematics.