Abstract
Let R be an Artinian ring, not necessarily with a unit, and let Ro be the group of all invertible elements of R with respect to the operation a o b = a + b + ab. We prove that the group Ro is a nilpotent group if and only if it is an Engel group and the quotient ring of the ring R by its Jacobson radical is commutative. In particular, Ro is nilpotent if it is a weakly nilpotent group or an n-Engel group for some positive integer n. We also establish that the ring R is strictly Lie-nilpotent if and only if it is an Engel ring and the quotient ring of the ring R by its Jacobson radical is commutative.
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