Bijective relative Gabriel correspondence over rings with torsion theoretic Krull dimension

(1)This work was started during the stay of the first author at the Univers ity of Wisconsin-Milwaukee as a visiting professor in the academic year 199 7/1998 and continued while he was a visiting scientist at the University of Manitoba, Winnipeg, Canada, March 1998, and a visiting scientist, with fin ancial support of the Alexander von Humboldt Foundation, at the University of Bielefeld, Germany, August-September 1998, and during his stay at The Oh io State University, Columbus, and the University of California, Santa Barb ara, as a visiting professor in the academic year 1998/1999. In the final s tage of its preparation he was partially supported by Grant 7D/2000 awarded by the Consiliul National al Cercetarii Stiintifice din Invatamantul Super ior, Romania. He thanks all these institutions for their hospitality and fi nancial support. (2)The second author gratefully acknowledges support from Grant OGP0007261 awarded by the Natural Sciences and Engineering Research Council of Canada. and Nastasescu (1981, Comm. Algebra 9, 1395-1426) give information about r ings that have Krull dimension or are noetherian relative to a torsion theo ry. The aim of this paper is to extend these results to rings R having rela tive Krull dimension with respect to a hereditary torsion theory tau on Mod -R such that any tau -torsion-free right R-module M has nonempty assassinat or. Since any ideal invariant hereditary torsion theory has this property i n view of a recent result by the authors (2000, J. Algebra 229, 498-513), t hese results apply in particular to the commutative case. (C) 2001 Academic Press.