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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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