Non-commutative Fitting invariants and annihilation of class groups

One can associate to each finitely presented module M over a commutative ring R an R-ideal FittR(M) which is called the (zeroth) Fitting ideal of M over R and which is an important natural invariant of M. We generalize this notion to o-orders in separable algebras, where o is a complete commutative noetherian local ring. As an application we construct annihilators of class groups assuming the validity of the Equivariant Tamagawa Number Conjecture for a certain motive attached to a Galois CM-extension of number fields.