Functor localization S-1() and Morita theory of category equivalences

Abstract

The main results we show in this paper are: • if A is a ring, S a saturated multiplicative set of A which satisfies the left Ore conditions then the category S−1A−Mod of the left S−1A-modules is equivalent to the category of left A-modules S-divisible and S-torsion free (denoted by STDA−Mod). • If A and B are Morita equivalent rings (denoted A ≈ B) and if S is a central saturated multiplicative set of A then the rings S−1A and S−1B are Morita equivalent and for any (two-sided) ideal I of A, if J is the ideal of B corresponding to I, then S−1I is an ideal of S−1A corresponding to the ideal S−1J of S−1B. • IfA ≈ B then the categories of complexes Comp(A−Mod) and Comp(B −Mod) are equivalent ( Comp(A−Mod) ≈ Comp(B −Mod) for short) and for all central multiplicative set S of A, 138 O. D. Diallo, M. B. Maaouia and M. Sanghare Comp ( S−1A−Mod ) ≈ Comp ( S−1B −Mod )