Notes on Whitehead space of an algebra

Abstract

LetRbe a ring, and denote by[R,R]the group generated additively by the additive commutators ofR. WhenRn=Mn(R)(the ring ofn×nmatrices overR), it is shown that[Rn,Rn]is the kernel of the regular trace function modulo[R,R]. Then consideringRas a simple left ArtinianF-central algebra which is algebraic overFwithChar F=0, it is shown thatRcan decompose over[R,R], asR=Fx+[R,R], for a fixed elementx∈R. The spaceR/[R,R]overFis known as the Whitehead space ofR. WhenRis a semisimple centralF-algebra, the dimension of its Whitehead space reveals the number of simple components ofR. More precisely, we show that whenRis algebraic overFandChar F=0, then the number of simple components ofRis greater than or equal todimF  R/[R,R], and whenRis finite dimensional overFor is locally finite overFin the case ofChar F=0, then the number of simple components ofRis equal todimF  R/[R,R].