International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 8, Pages 509-512
Notes on Whitehead space of an algebra
Department of Mathematics, University of Zanjan, Zanjan, Iran
Received 2 August 2001
Copyright © 2002 M. Arian-Nejad. 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.
Let be a ring, and denote by the group generated additively by the additive commutators of . When (the ring of matrices over ), it is shown that is the kernel of the regular trace function modulo
. Then considering as a simple left Artinian -central algebra which is algebraic over with , it is shown that can decompose over , as , for a fixed element . The space over is known as the Whitehead space of . When is a semisimple central -algebra, the dimension of its Whitehead space reveals
the number of simple components of . More precisely, we show that when is algebraic over and , then the number of simple components of is greater than or equal to , and when is finite dimensional over or is locally finite over in the case of , then the number of simple components of is equal to .