Wakamatsu tilting modules with finite injective dimension

Let R be a left Noetherian ring, S a right Noetherian ring and (R) omega a Wakamatsu tilting module with S = End( (R) omega). We introduce the notion of the omega-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n a (c) 3/4 0, we prove that l.id (R) (omega) = r.id (S) (omega) a (c) 1/2 n if and only if every finitely generated left R-module and every finitely generated right S-module have omega-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.