




Volume 36 • Number 2 • 2013 

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Weak Annihilator Property of MalcevNeumann Rings
Ouyang Lunqun and Liu Jinwang
Abstract.
Let $R$ be an associative ring with identity, $G$ an totally ordered group, $\sigma$ a map from $G$ into the group of automorphisms of $R$, and $t$ a map from $G\times G$ to the group of invertible elements of $R$. The weak annihilator property of the MalcevNeumann ring $R\ast((G))$ is investigated in this paper. Let $\text{nil}(R)$ denote the set of all nilpotent elements of $R$, and for a nonempty subset $X$ of a ring $R$, let $N_R(X)=\{a\in R\mid Xa\subseteq \text{nil}(R)\}$ denote the weak annihilator of $X$ in $R$. Under the conditions that $R$ is an $NI$ ring with $\text{nil}(R)$ nilpotent and $\sigma$ is compatible, we show that: (1) If the weak annihilator of each nonempty subset of $R$ which is not contained in $\text{nil}(R)$ is generated as a right ideal by a nilpotent element, then the weak annihilator of each nonempty subset of $R\ast((G))$ which is not contained in $\text{nil}(R\ast((G)))$ 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 $R\ast((G)))$ is generated as a right ideal by a nilpotent element. As a generalization of left APPrings, we next introduce the notion of weak APPrings and give a necessary and sufficient condition under which the ring $R\ast((G))$ over a weak APPring $R$ is weak APP.
2010 Mathematics Subject Classification: 16W60
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