Localization in general rings, a historical survey

Abstract

Introduction The process of introducing fractions in a ring, or localization, has been applied in many different ways in algebra and geometry, and more recently it has also been used for noncommutative rings. Our object here is to survey the different methods of forming fractions, with particular emphasis on the noncommutative case. After a statement of the problem in §2 we look in §3 at different classes of rings that permit the introduction of fractions but are not embeddable in skew fields, and in §4 describe some topological methods. §5 deals with fractions in a general ring, including a statement of the necessary and sufficient conditions for embeddability in a skew field. Various classes of rings are considered in §6 and specific examples of such rings are given in §7. I should like to thank George Bergman, whose careful reading provided comments which resulted in a number of improvements. I am also indebted to a referee whose comments helped to clarify the text. Throughout, all rings are associative, with a unit element, denoted by 1, which is inherited by subrings, preserved by homomorphisms and which acts unitally on modules. If 1 ≠ 0 and every non-zero element has an inverse, we speak of a skew field , but we shall frequently omit the prefix “skew”, so that a “field” will mean a not necessarily commutative division ring.