Noncommutative Fitting invariants and improved annihilation results

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 named author generalised this notion by replacing R with a (not necessarily commutative) o-order Λ 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) Λ-module M by a certain ideal H(Λ) of the centre of Λ. In contrast to the commutative case, this ideal can be properly contained in the centre of Λ. In the present article, we determine explicit lower bounds for H(Λ) in many cases. Furthermore, we define a class of ‘nice’ orders Λ 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(Λ) being the best possible.