Inversive localization in noetherian rings

Now the process of forming fractions has been generalized to non-commutative rings in a number of ways. The simplest is Ore’s method (see e.g. [ Z ] ) , but this is of limited applicability. Another method, much studied lately, depends on forming injective hulls; it is usually expressed within the framework of torsion theories (cf. [6], [7], [l I]). It leads to a generalized quotient ring which reduces to Ore’s construction whenever the latter is applicable, and like the latter it is not left-right symmetric. We shall call this the injective method. A second way of generalizing Ore’s method is to invert matrices rather than elements. This allows one to obtain an explicit form for the quotient ring produced; unlike the injective method it leads to actual inverses and is left-right symmetric, but in general it is more difficult to determine the kernel of the canonical mapping. This may be called the inversive method; it is described in [3]. If instead of inverting matrices, we merely make them right invertible, we obtain an intermediate method, which may be called semi-inversive. I t is natural to try to apply these methods to obtain a non-commutative localization process. For a Noetherian (semi)prime ring, Ore’s method can always be applied to yield a (semi)simple Artinian quotient ring (Goldie’s theorem), but this is no longer so when we try to localize a t a prime ideal of a Noetherian ring. A number of ways of performing such a localization have 679 IV).