International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 2, Pages 247-253
Rings with involution whose symmetric elements are central
Department of Actuarial Mathematics, University of Manitoba, Winnipeg R3T 2N2, Manitoba, Canada
Received 30 May 1978; Revised 6 February 1979
Copyright © 1980 Taw Pin Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In a ring with involution whose symmetric elements are central, the skew-symmetric elements form a Lie algebra over the commutative ring . The classification of such rings which are -torsion free is equivalent to the classification of Lie algebras over equipped with a bilinear form that is symmetric, invariant and satisfies . If is a field of char , and then is a semisimple Lie algebra if and only if is nondegenerate. Moreover, the derived algebra is either the pure quaternions over or a direct sum of mutually orthogonal abelian Lie ideals of .