### Rings having solvable adjoint groups

P. B. Bhattacharya, S. K. Jain
1970 Proceedings of the American Mathematical Society
Let °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 °R is finitely generated solvable group then R is finite and hence °R =P\ o?io • • • oPm where Pi are pairwise commuting ^-groups. (2) Let R be a locally matrix ring or a prime ring with nonzero socle. Then °R is solvable iff R is either a field or a 2X2 matrix ring over a field having at most 3 elements. For a ring R let JiR)
more » ... te the Jacobson radical, °R the group of quasi-regular elements with respect to circle operation and C/(i?) the group of units if R has identity. We know that if R has identity, then °R is isomorphic to UiR). °R is called the adjoint group of R. The object of this paper is to study certain classes of rings R for which°R is nilpotent, supersolvable or solvable. 1.1. Let M be a unital free i?-module over a ring R in which 2 is invertible and let c7 (5) be supersolvable where 5, = Hom«(M, M). Then S = R. If S^R, then 5 contains a copy T of a 2X2 matrix ring over R. In this case we choose a, b, cE U{T) such that -GO--CO--C!D-Direct computation yields a~1b~1ab = a, b~1c~1bc = b. These relations first imply that a, b belong to the derived group of U{T) and further the relation a~1b~1ab = a implies that the derived group cannot be nilpotent. Hence UiT) cannot be supersolvable in contradiction to the hypothesis that l7(5) is supersolvable. This proves 1.1. The following example shows that if 2 =0 in a ring R then 1.1 may not be true: The group of units of a 2 X2 matrix ring over a field of 2 elements is supersolvable. 1.2. Let M be a unital free module over a ring R of characteristic 2 and let the group UiS) be nilpotent where S = HomRiM, M). Then S = R.