Preface

Summary

The finite dimensional representations of a ring over commutative fields have been studied in great detail for many types of ring, for example, group rings or the enveloping algebras of finite dimensional Lie algebras, but little is known about the finite dimensional representations of a ring over skew fields although such information might be of great use. The first part of this book is devoted to a classification of all possible finite dimensional representations of an arbitrary ring over skew fields in terms of simple linear data on the category of finitely presented modules over the ring. The second part is devoted to a fairly detailed study of those skew fields that arise in the first part and in the work of Cohn on firs and skew fields.

As has been said, the main goal at the beginning is to study finite dimensional representations of a ring over skew fields. An alternative view of this is that we should like to classify all possible homomorphisms from a ring to simple artinian rings; such a study was carried out in the case of one dimensional representations which are simply homomorphisms to skew fields by Cohn who showed that these homomorphisms are determined by which sets of matrices become zero-divisors over the skew field and gave a characterisation of the sets of matrices that could be exactly those that become singular under a homomorphism to a skew field. This theory has a particular application to firs, rings such that every left and right ideal are free of unique rank to show that they have universal homomorphisms to skew fields.