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.