Notes on Whitehead space of an algebra

Let R be a ring, and denote by [ R , R ] the group generated additively by the additive commutators of R . When R n = M n ( R ) (the ring of n × n matrices over R ), it is shown that [ R n , R n ] 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 = F x + [ 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 dim F 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 dim F R / [ R , R ] .