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.
