In Comm. Algebra 30(3) (2002), 1475–1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act A S over a monoid S that are based on the extent to which the functor A S ⊗ — preserves equalizers. The present paper discusses in detail one of these notions, annihilator-flatness. For s,t ∈ S , the set L s,t = { u ∈ S: us = ut }, if it is non-empty, is a left ideal of S , called an annihilator ideal, and an act A S is called annihilator-flat if the natural mapping A ⊗ L s,t → A, a ⊗ u ↦ au, is injective, for each s,t ∈ S. Every weakly flat right act is obviously annihilator-flat, but the converse is by no means true. A monoid is called right absolutely annihilator-flat (r.a.a-f.) if all of its right acts are annihilator-flat. Left annihilator-flatness is defined dually. In this paper, we present a description of right absolute annihilator-flatness that is intrinsic to S (that is, that makes no reference to tensor products or to S -acts). This result is then employed to show that all full transformation monoids are both left and right absolutely annihilator-flat. (In contrast, it is known that T X is weakly right absolutely flat if and only if X is a singleton and weakly left absolutely flat if and only if X is finite.) A structural characterization is given of the r.a.a-f. completely (0-) simple semigroups and of the r.a.a-f. normal bands (with 1 adjoined). The r.a.a-f. commutative monoids are shown to be exactly those in which every principal ideal is pullback-flat.