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International Journal of Mathematics and Mathematical Sciences
Volume 31, Issue 8, Pages 509-512
http://dx.doi.org/10.1155/S0161171202007998

Notes on Whitehead space of an algebra

Department of Mathematics, University of Zanjan, Zanjan, Iran

Received 2 August 2001

Copyright © 2002 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.

Abstract

Let R be a ring, and denote by [R,R] the group generated additively by the additive commutators of R. When Rn=Mn(R) (the ring of n×n matrices over R), it is shown that [Rn,Rn] is the kernel of the regular trace function modulo [R,R]. Then considering R as a simple left Artinian F-central algebra which is algebraic over F with CharF=0, it is shown that R can decompose over [R,R], as R=Fx+[R,R], for a fixed element xR. The space R/[R,R] over F is known as the Whitehead space of R. When R is a semisimple central F-algebra, the dimension of its Whitehead space reveals the number of simple components of R. More precisely, we show that when R is algebraic over F and CharF=0, then the number of simple components of R is greater than or equal to dimFR/[R,R], and when R is finite dimensional over F or is locally finite over F in the case of CharF=0, then the number of simple components of R is equal to dimFR/[R,R].