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To each finitely presented module M over a commutative ring R one can associate an R-ideal Fit R (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.

Let E be a number field and G be a finite group. Let A be any O E -order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] ∼ = χ M χ is explicitly computable and each M χ is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α 1 , . . . , α d ∈ X such that X = Aα 1 ⊕ · · · ⊕ Aα d or determines that no such elements exist.Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O L , the ring of algebraic integers of L, and A to be the associated order A(E [G]; O L ) ⊆ E [G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = Q.

Abstract. Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a prime and let r ≤ 0 be an integer. By examining the structure of the p-adic group ring Z p [G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h 0 (Spec(L))(r), Z[G]). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a large class of interesting extensions, including cases in which the full ETNC is not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.

Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, this result does not depend on the vanishing of the relevant Iwasawa µ-invariant.