MMN-1531

\item[$(a)$] if the adjoint group $R^{\circ}$ of a Jacobson radical ring $R$ is an $FC$-group, then $R$ is commutative or two-sided $T$-nilpotent (and so $R^{\circ}$ is a hypercentral group), \item[$(b)$] the index $|R:Z(R)$ is finite if and only if the set of all inner derivations $\IDer R$ is finite (and then $R$ contains a central ideal $I$ of finite index such that $I\cdot C(R)=0$), \item[$(c)$] if a Jacobson radical ring $R$ has the adjoint group $R^{\circ}$ with a finite number of conjugacy classes, then $$R=R_{p_1}\oplus \cdots \oplus R_{p_t}\oplus D$$ is a ring direct sum of Jacobson radical rings $R_{p_i}$ and $D$, where $D^+$ is a torsion-free divisible group, $D^{\circ}$ is a group with a finite number of conjugacy classes, $R_{p_i}^+$ is a finite $p_i$-group $(i=1,\ldots ,t)$ and $p_1,\ldots ,p_t$ are pairwise distinct primes, \item[$(d)$] a Jacobson radical ring $R$ with two conjugacy classes is either a nilpotent ring that contains only two elements or a simple domain with the torsion-free divisible additive group $R^+$ and the torsion-free simple adjoint group $R^{\circ}$. \end{itemize} We also characterize left Artinian rings with the finite set of all derivations $\Der R$ (respectively inner derivations $\IDer R$).

Vol. 18 (2017), No. 2, pp. 623-637

DOI: 10.18514/MMN.2017.1531

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