On Artinian rings satisfying the Engel condition - Evstaf’ev, R. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
Let $R$ be an Artinian ring, not necessarily with a unit element, and let $R^{°}$ be the group of all invertible elements of $R$ under the operation $a ° b = a + b + ab.$ We prove that $R^{°}$ 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 $R^{°}$ 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 $R^{°}$ — група оборотних елементів кільця $R$ відносно операції $a ° b = a + b + ab.$