2019
Том 71
№ 11

# Evstaf’ev R. Yu.

Articles: 2
Brief Communications (Russian)

### On Artinian rings satisfying the Engel condition

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1264–1270

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

Brief Communications (Ukrainian)

### Artinian rings with nilpotent adjoint group

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 417–426

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.