4 - Localization

Summary

This chapter deals with the formation of fractions in general rings. In the commutative case a necessary and sufficient condition for the existence of a field of fractions is the absence of zero-divisors (and the condition 1 ≠ 0), and the construction as fractions ab^-1 is well known. As we saw in 1.3, the same method of construction still applies in Ore domains, though the verification is a little more involved. In the general case the difficulties are both theoretical – the criterion for embeddability is quite complicated and cannot be stated as an elementary sentence – and practical – a sum of fractions cannot generally be brought to a common denominator. The practical problem is overcome by inverting matrices rather than elements. After some general remarks on epimorphisms and localizations in 4.1, we go on to show in 4.2 that all elements of the field of fractions (if one exists) can be found by solving matrix equations, and something like a normal form (in the case of firs) is presented in 4.7. On the theoretical side we shall meet a criterion for a ring to possess a field of fractions in Th. 4.5, but what turns out to be more useful is a sufficient condition for a ring to have a universal field of fractions (Th. 5.3); the latter, when it exists, is unique up to isomorphism, unlike a field of fractions, of which there may be many, e.g. for a free algebra.