Weak Annihilator Property of Malcev-Neumann Rings

Let R be an associative ring with identity, G an totally ordered group, σ a map from G into the group of automorphisms of R, and t a map from G x G to the group of invertible elements of R. The weak annihilator property of the Malcev-Neumann ring R*((G)) is investigated in this paper. Let nil(R) denote the set of all nilpotent elements of R, and for a nonempty subset X of a ring R, let NR(X) = {a element of R | Xa subseteq nil(R)} denote the weak annihilator of X in R. Under the conditions that R is an NI ring with nil(R) nilpotent and sigma is compatible, we show that: (1) If the weak annihilator of each nonempty subset of R which is not contained in nil(R) is generated as a right ideal by a nilpotent element, then the weak annihilator of each nonempty subset of R *((G)) which is not contained in nil(R*((G))) is generated as a right ideal by a nilpotent element. (2) If the weak annihilator of each nonnilpotent element of R is generated as a right ideal by a nilpotent element, then the weak annihilator of each nonnilpotent element of R* ((G))) is generated as a right ideal by a nilpotent element. As a generalization of left APP-rings, we next introduce the notion of weak APP-rings and give a necessary and sufficient condition under which the ring R*((G)) over a weak APP-ring R is weak APP.