The group ring K[G] is an associative ring which exhibits properties of the group G and the field of coefficients K. As the name implies, its study is a meeting place for group theory and ring theory, and as such it has been approached from many different points of view. For example, the finite group theorist does character theory and the analyst does operator theory and Fourier analysis. On the other hand, our approach here is purely algebraic and ring theoretic.
This algebraic study of group rings was initiated in 1949 by I. Kaplansky, but it did not really catch on until the fundamental work of S. A. Amitsur appeared some ten years later. Since then the subject has been pursued by a small but growing number of researchers, and at this point in time it has reached the stage in its development where a coherent account of the basic results is needed. Such is the goal of this text, and a rough outline of the topics covered is given below.
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