International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1980 / Article

Open Access

Volume 3 |Article ID 762359 | https://doi.org/10.1155/S0161171280000178

Taw Pin Lim, "Rings with involution whose symmetric elements are central", International Journal of Mathematics and Mathematical Sciences, vol. 3, Article ID 762359, 7 pages, 1980. https://doi.org/10.1155/S0161171280000178

Rings with involution whose symmetric elements are central

Received30 May 1978
Revised06 Feb 1979

Abstract

In a ring R with involution whose symmetric elements S are central, the skew-symmetric elements K form a Lie algebra over the commutative ring S. The classification of such rings which are 2-torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f that is symmetric, invariant and satisfies [[x,y],z]=f(y,z)xf(z,x)y. If S is a field of char 2, f0 and dimK>1 then K is a semisimple Lie algebra if and only if f is nondegenerate. Moreover, the derived algebra K is either the pure quaternions over S or a direct sum of mutually orthogonal abelian Lie ideals of dim2.

Copyright © 1980 Hindawi Publishing Corporation. 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.


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