An element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.

中文翻译：

---

歼灭者-稳定性和尼科尔森的两个问题

一个元素一种在一圈R是左歼灭者稳定的（或左作为) 如果，无论何时$Ra+{\rm l}(b)=R$和$b\in R$,$au\in {\rm l}(b)$对于一个单位你在R, 和环R是左作为如果它的每个元素都留下，则响铃作为. 在本文中，我们证明左作为环中的元素形成一个乘法闭集，对 Nicholson 的问题给出肯定的回答 [J.纯应用。阿尔格。221(2017), 2557–2572.]。该结果用于获得形式三角矩阵环的充分充要条件作为. 作为应用程序，我们提供了 left 的示例作为戒指R三角矩阵在其上环${\mathbb T}_n(R)$没有留下作为对所有人$n\ge 2$. 这些例子对尼科尔森的另一个问题给出了否定的回答[J.纯应用。阿尔格。221(2017), 2557–2572.] 是否R/J(R)被留下作为暗示R离开了作为.