Abstract
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/I n the nilradical of A/ I n and a sequence Q n +1 → Q n of surjective ring homomorphisms with inverse limit Q a semiperfect ring.
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Mcconnell, M., Sandomierski, F.L. (1997). Localization in Noetherian Rings. In: Jain, S.K., Rizvi, S.T. (eds) Advances in Ring Theory. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1978-1_20
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