To each finitely presented module M over a commutative ring R one can associate an R-ideal FitR(M) which is called the (zeroth) Fitting ideal of M over R and which is always contained in the R-annihilator of M. In an earlier article, the second author generalised this notion by replacing R with a (not necessarily commutative) o-order in a finite dimensional separable algebra, where o is an integrally closed complete
commutative noetherian local domain. To obtain annihilators, one has to multiply the Fitting invariant of a (left) �-module M by a certain ideal H(�) of the centre of �. In
contrast to the commutative case, this ideal can be properly contained in the centre of �. In the present article, we determine explicit lower bounds for H(�) in many cases.
Furthermore, we define a class of `nice' orders � over which Fitting invariants have several useful properties such as good behaviour with respect to direct sums of modules,
computability in a certain sense, and H(�) being the best possible.