Rings Having Solvable Adjoint Groups

Abstract

Let $^\circ R$ denote the group of quasi-regular elements of a ring $R$ with respect to circle operation. The following results have been proved: (1) If $R$ is a perfect ring and $^\circ R$ is finitely generated solvable group then $R$ is finite and hence $^\circ R = {P_1} \circ {P_2} \circ \; \cdots \; \circ {P_m}$ where ${P_i}$ are pairwise commuting $p$-groups. (2) Let $R$ be a locally matrix ring or a prime ring with nonzero socle. Then $\circ R$ is solvable iff $R$ is either a field or a $2 \times 2$ matrix ring over a field having at most $3$ elements.