Notes on Whitehead space of an algebra

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 Char F=0, it is shown that R can decompose over [R,R], as R=Fx+[R,R], for a fixed element x∈R. 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 Char F=0, then the number of simple components of R is greater than or equal to dimF R/[R,R], and when R is finite dimensional over F or is locally finite over F in the case of Char F=0, then the number of simple components of R is equal to dimF R/[R,R].