On homological dimensions of simple modules over non-commutative rings

It is known that for a simple module S over a commutative ring R, fd(R)(S) = id(R)(S). Let R, T be commutative rings and R --> T a ring homomorphism, if T is a Noetherian ring and self-injective, then fd(R)(T) = id(R)(T). In this paper we use the equalities of mixed functors to generalize these results over non-commutative rings.