Notes on Whitehead space of an algebra. (Q700897)

Summary: 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 \(n\times n\) 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 \(\text{Char\,}F=0\), it is shown that \(R\) can decompose over \([R,R]\), as \(R=Fx+[R,R]\), for a fixed element \(x\in R\). 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 \(\text{Char\,}F=0\), then the number of simple components of \(R\) is greater than or equal to \(\dim_FR/[R,R]\), and when \(R\) is finite dimensional over \(F\) or is locally finite over \(F\) in the case of \(\text{Char\,}F=0\), then the number of simple components of \(R\) is equal to \(\dim_FR/[R,R]\).