Vol. 44, No. 1, 1973

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A generalization of a theorem of Jacobson. II

Susan Montgomery

Vol. 44 (1973), No. 1, 233–240

According to a well-known theorem of Jacobson, a ring R in which xn(x) = x ( n(x) an integer > 1) for each x in R must be commutative. This paper completes the description of rings with involution in which the above condition is imposed only on the symmetric elements. It is shown that in any such ring, the Jacobson radical J(R) is nilpotent of index 3, and RlJ(R) is a subdirect sum of fields and 2 ×2 matrix rings. This had been shown previously under the assumption that R was an algebra over a field of characteristic not 2. In addition, it is shown that such a ring of characteristic 2 must actually be commutative. These results are best possible, since if R is 2 torsion free, R need not be commutative unless R is a division ring. Finally, using these methods, a conjecture of Jacobson on restricted Lie algebras is confirmed in a special case.

Mathematical Subject Classification
Primary: 16A28
Secondary: 16A68
Received: 27 September 1971
Revised: 29 October 1971
Published: 1 January 1973
Susan Montgomery
Department of Mathematics
University of Southern California
Los Angeles CA 90265-2532
United States