Injective modules and classical localization in Noetherian rings

One of the main problems in the growing theory of noncommutative Noetherian rings can be loosely stated thus; If p is a prime ideal of a Noetherian ring R, what should one mean by the localization Rv of R at p? When does Rv exist and when is it nice? This problem has been considered by Goldie [1] and by Lambek and Michler [5]. In this note, we indicate a new approach to this problem and some of its advantages. We also introduce the concept of a left exact biradical for a ring, which may be of independent interest. Details will appear elsewhere. As usual, a ring is Noetherian if it has the ascending chain condition on right ideals as well as left ideals. A subset of a ring is an Ore set if it is right Ore as well as left Ore. We refer the reader to [9] for all unexplained terminology and results concerning left exact radicals. Let R be a ring. The complete lattice of all left exact radicals for mod-R (resp. K-mod) is denoted as Kr (resp. Kt). If 2 is a multiplicatively closed subset of R, p@eKr and X2}sKl are defined as follows: For each Memod-R (resp. MejR-mod), p®(M) (resp. A^M)) is the largest submodule of M, each element of which is annihilated by some element of Q). If a is an ideal of R, we define p* as sup{p e Kr\p(R/a) = 0} and A* as sup{A e Kt\)i(R/a) = 0}. The multiplicatively closed set {reR\[r + a] is regular in R/a} is denoted as ^(ct).