On Artinian rings satisfying the Engel condition

  • R. Yu. Evstaf’ev


Let $R$ be an Artinian ring, not necessarily with a unit element, and let $R^{\circ}$ be the group of all invertible elements of $R$ under the operation $a \circ b = a + b + ab.$ We prove that $R^{\circ}$ is a nilpotent group if and only if it is an Engel group and the ring $R$ modulo its Jacobson radical is commutative. In particular, the group $R^{\circ}$ is nilpotent if it is weakly nilpotent or $n$-Engel for some positive integer $n$. We also establish that $R$ is a strictly Lie-nilpotent ring if and only if R is an Engel ring and $R$ modulo its Jacobson radical is commutative.
Нехай $R$ — артінове кільце, необов'язково з одиницею, i $R^{\circ}$ — група оборотних елементів кільця $R$ відносно операції $a \circ b = a + b + ab.$
How to Cite
Evstaf’ev, R. Y. “On Artinian Rings Satisfying the Engel Condition”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 9, Sept. 2006, pp. 1264–1270, https://umj.imath.kiev.ua/index.php/umj/article/view/3527.
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