ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON
Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 258-265
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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.
XIA, GUOLI; ZHOU, YIQIANG. ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON. Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 258-265. doi: 10.1017/S0017089520000154
@article{10_1017_S0017089520000154,
author = {XIA, GUOLI and ZHOU, YIQIANG},
title = {ANNIHILATOR-STABILITY {AND} {TWO} {QUESTIONS} {OF} {NICHOLSON}},
journal = {Glasgow mathematical journal},
pages = {258--265},
year = {2021},
volume = {63},
number = {2},
doi = {10.1017/S0017089520000154},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000154/}
}
TY - JOUR AU - XIA, GUOLI AU - ZHOU, YIQIANG TI - ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON JO - Glasgow mathematical journal PY - 2021 SP - 258 EP - 265 VL - 63 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000154/ DO - 10.1017/S0017089520000154 ID - 10_1017_S0017089520000154 ER -
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