Artinian rings with nilpotent adjoint group

Abstract

Let $R$ be an Artinian ring (not necessarily with unit element), let $Z(R)$ be its center, and let $R ^{\circ}$ be the group of invertible elements of the ring $R$ with respect to the operation $a ∘ b = a + b + ab$. We prove that the adjoint group $R ^{\circ}$ is nilpotent and the set $Z (R) + R ^{\circ}$ generates $R$ as a ring if and only if $R$ is the direct sum of finitely many ideals each of which is either a nilpotent ring or a local ring with nilpotent multiplicative group.