Preface

Summary

It is not your duty to complete the work,

But neither are you free to desist from it.

R. Tarphon, Sayings of the Fathers.

One of the questions that intrigued me in the 1950s was to find conditions for an embedding of a non-commutative ring in a skew field to be possible. I felt that such an embedding should exist for a free product of skew fields, but there seemed no obvious route. My search eventually led to the notion of a free ideal ring, fir for short, and I was able to prove (i) the free product of skew fields (amalgamating a skew subfield) is a fir and (ii) every fir is embeddable in a skew field. Firs may be regarded as the natural generalization (in the non-commutative case) of principal domains, to which they reduce when commutativity is imposed. The proof of (i) involved an algorithm, which when stated in simple terms, resembled the Euclidean algorithm but depended on a condition of linear dependence. In this form it could be used to characterize free associative algebras, and this ‘weak’ algorithm enables one to develop a theory of free algebras similar to that of a polynomial ring in one variable. Of course free algebras are a special case of firs, and other facts about firs came to light, which were set forth in my book Free Rings and their Relations (a pun and a paradox). It appeared in 1971 and in a second edition in 1985. A Russian translation appeared in 1975.

More recently there has been a surprising increase of interest, in many fields of mathematics, in non-commutative theories.