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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Rings having solvable adjoint groups

Authors: P. B. Bhattacharya and S. K. Jain
Journal: Proc. Amer. Math. Soc. 25 (1970), 563-565
MSC: Primary 16.50
MathSciNet review: 0271154
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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.

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Keywords: Perfect rings, locally matrix rings, prime rings with nonzero socles, solvable adjoint groups
Article copyright: © Copyright 1970 American Mathematical Society