Localization in Noetherian Rings

When A is a left Noetherian ring with nilradical N, then there is a unitary subring B of A and ∑ a left denominator set in B such that Q, the ring of left fractions of B with respect to ∑ is left Artinian. Furthermore, for P = Q ⊗ _B A, P is a flat right A-module of type FP such that M, a left A-module, is C(N)-torsion if and only if P ⊗ _A M = 0. For the functors T = P ⊗ _A (·): A mоd→ Q-mod and S = Hom _Q (P,·): Q-mod→ A-mod, the natural transformation 1 → ST, M ↦ ST(M) is the localization of M in A-mod with respect to the torsion theory on A-mod corresponding to the multiplicative set C(N).When I is a semiprime ideal of a left Noetherian ring, then for each positive integer n, a ring Q _n is constructed as above for N = I/In the nilradical of A/In and a sequence Q _{n +1} → Q _n of surjective ring homomorphisms with inverse limit Q a semiperfect ring.