ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON.

An element a in a ring R is left annihilator-stable (or left AS) if, whenever Ra + l(b) = R with b ∈ R, a−u ∈ 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 T_n(R) are not left AS for all n ≥ 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. [ABSTRACT FROM AUTHOR]