2017
Том 69
№ 6

All Issues

On Artinian rings satisfying the Engel condition

Evstaf’ev R. Yu.

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Abstract

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 $a \circ b = a + b + ab.$

English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 9, pp 1433-1440.

Citation Example: Evstaf’ev R. Yu. On Artinian rings satisfying the Engel condition // Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1264–1270.

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