Abstract

This paper will briefly survey some recent methods of localization in group rings, which work in more general contexts than the classical Ore localization. In particular the Cohn localization using matrices will be described, but other methods will also be considered.

Introduction

Let R be a commutative ring and let S = {s ∈ R | sr ≠ 0 for all r ∈ R \ 0}, the set of non-zerodivisors of R. Then, in the same manner as one constructs ℚ from ℤ, we can form the quotient ring RS–1 which consists of elements of the form r/s with r ∈ R and s ∈ S, and in which r1/s1 = r2/s2 if and only if r1s2 = s1r2. We can consider R as a subring RS–1 by identifying r ∈ R with r/1 ∈ RS–1. Then RS–1 is a ring containing R with the property that every element is either a zerodivisor or invertible. Furthermore, every element of RS–1 can be written in the form rs–1 with r ∈ R and s ∈ S (though not uniquely so). In the case R is an integral domain, then RS–1 will be a field and will be generated as a field by R (i.e. if K is a subfield of RS–1 containing R, then K = RS–1). Moreover if K is another field containing R which is generated by R, then K is isomorphic to RS–1 and in fact there is a ring isomorphism RS–1 → K which is the identity on R.