Abstract
AbstractAn 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.
Publisher
Cambridge University Press (CUP)
Reference15 articles.
1. 8. Horoub, A. , $\mathcal L$ -stability in rings and left Quasi-duo rings, PhD Thesis (The University of Calgary, Canada, 2018).
2. Rings with internal cancellation
3. 14. Nicholson, W. K. , Corrigendum to “Annihilator-stability and unique generation” [J. Pure Appl. Algebra 221 (2017) 2557–2572], J. Pure Appl. Alg. 222 (2018), 3334–3335.
4. Factorization in Commutative Rings with Zero Divisors
5. A criterion for unit-regularity
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