Injective modules and localization in noncommutative Noetherian rings

Let 2 be a semiprime ideal in a right Noetherian ring R and W) = {c e R|[c + A] regular in R/Mi. We investigate the following two conditions: (A)C(?) is a right Ore set in R. (B) (!$) is a right Ore set in R and the right ideals of R,, the classical right quotient ring of R w.r.t. e(B), are closed in the J(R,)-adic topology. The main results show that conditions (A) and (B) can be characterized in terms of the injective hull of the right R-module R/9. The J-adic completion of a semilocal right Noetherian ring is also considered. Introduction. The well-known work of Matlis [12] makes a striking use of the localization and the completion to describe the injective modules over commutative Noetherian rings. It led us to examine the interaction between the injective modules and the localization in the context of noncommutative Noetherian rings. Let 2 be a semiprime ideal in a right Noetherian ring R. Goldie's work in [2] shows that, in any attempt at defining the localization of R at 2, the set e(2) = Ec e R|[c + !1 regular in R/Mi deserves special attention. In this connection, of particular interest is the condition (A): Q(!) is a right Ore set in R. For, if (A) is satisfied then the usual procedure of clearing off the zero-divisors works and the classical right quotient ring R. of R w.r.t. (!(!2) can be constructed. Some preliminary results on the classical right quotient rings occupy ?1. The main result of ?2 (Theorem 2.1) shows that if (A) is satisfied then R, is a very tame ring indeed. It is semilocal; its Jacobson radical is the right ideal generated by the image of !; further, ER(R/0) ER (R,/J(R, )). It is rather surprising that these results were known only under some severe conditions on R. It is then natural to seek some useful characterizations of the condition (A). Lambek and Michler have shown (Theorem 5.6 of [10]) that (A) is satisfied if and Received by the editors November 10, 1972. AMS (MOS) subject classifications (1970). Primary 16A08, 16A46, 16A52; Secondary 16A10.