Bijective Relative Gabriel Correspondence over Rings with Torsion Theoretic Krull Dimension

A series of results by Asensio and Torrecillas (1992, Comm. Algebra20, 847–866), Gordon and Robson (1973, “Krull Dimension,” Memoirs of the American Mathematical Society, Vol. 133), Kim and Krause (1999, Comm. Algebra27, 3339–3351), and N st sescu (1981, Comm. Algebra9, 1395–1426) give information about rings that have Krull dimension or are noetherian relative to a torsion theory. The aim of this paper is to extend these results to rings R having relative Krull dimension with respect to a hereditary torsion theory τ on Mod-R such that any τ-torsion-free right R-module M has nonempty assassinator. Since any ideal invariant hereditary torsion theory has this property in view of a recent result by the authors (2000, J. Algebra229, 498–513), these results apply in particular to the commutative case.