Hereditary Torsion Theory Counter Equivalences Original Research Article

One of the main ingredients of a tilting theory, a generalization of Morita equivalence betweenR-Mod andS-Mod that has evolved from the work of S. Brenner and M. C. R. Butler (Lecture Notes in Math.,8321980, 103–170) and D. Happel and C. M. Ringel (Trans. Amer. Math. Soc.,2741982, 399–443), is a pair of category equivalences imagetwo left arrowsimage and imagetwo left arrowsimage between the members of torsion theories (image, image) inR-Mod and (image, image) inS-Mod. Recently, we began to investigate this phenomenon as a further generalization of Morita equivalence, in particular showing that such equivalences are induced by pairs of bimodules (R. R. Colby and K. R. Fuller,Comm. Algebra,231995, 4833–4849). Here we are concerned with conditions under which the torsion theories are hereditary. We begin with a characterization of the bimodules, and we apply it to determine just when one or both of the torsion theories are hereditary, to obtain examples, and to investigate global dimensions and Grothendieck groups ofRandS. We assume the notation and conventions of F. W. Anderson and K. R. Fuller (1992, “Rings and Categories of Modules,” 2nd ed., Springer-Verlag, New York) and B. Stenström (1970, “Rings of Quotients,” Springer-Verlag, New York/Heidelberg/Berlin). In addition all “subcategories” are full subcategories of modules and all “functors” are additive functors.