Global dimension of the endomorphism ring and *n-modules

We show that, if T is a selfsmall and selforthogonal module over a noetherian ring R of finite global dimension with the endomorphism ring A, then . Applying the result we give answers to two questions left in [J. Wei et al., J. Algebra 168 (2) (2003) 404–418] concerning basic properties of *n-modules, by showing that the flat dimension of a *n-module with n 3 over its endomorphism ring can even be arbitrarily far from the integer n while the flat dimension of a *2-module over its endomorphism ring is always bounded by the integer 2 and showing that *n-modules are not finitely generated in general, even in case n=2.