This is a substantially extended version of the notes on 25 lectures delivered at the Pennsylvania State University during Spring Term 1981 under 572.2 (“Special Topics in Algebra”).

Most of the material has been presented earlier elsewhere: in the Seminar Bielefeld-Göttingen during the Summer Term 1967 (mainly Part III), in lectures at the Universität Bielefeld during the Academic Year 1977/78 and in lectures at the Université de Grenoble in February/March 1979 (mainly §§19/20.). Some of the material has been discussed afterwards in the course of different lectures which I delivered at the Universität Bielefeld during the Academic Year 1981/82.

The text falls into three parts: Skew fields and simple rings, Skew fields and Brauer groups, and Reduced K1-theory of skew fields. As regards their contents the reader is advised to consult the introductory remarks at the beginning of each of these parts on pages 1, 57 and 125 respectively.

During the preparation of the final draft of these notes I have enjoyed the assistance of B. Fein (Oregon State University), D. Garbe (Universität Bielefeld), I.M. James (Oxford University, editor of the LMS Lecture Note Series), Ch. Preston (Universität Bielefeld), S. Rosset (Tel-Aviv University), B. Weisfeiler (Pennsylvania State University) and J. Tate (Harvard University, who communicated so far unpublished work to me and gave permission to publish his arguments here; cf. §11.); I am grateful for their help.