I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is:
A sufficient condition for the rank of a free module over a ring $R$ to be uniquely defined is the existence of a homomorphism $\phi:R \to k$ into a skew-field $k$. In this case the concept of the rank of a module can be extended to projective modules as follows. The homomorphism $\phi$ induces a homomorphism of the groups of projective classes $\phi:K_0 R \to K_0 k \approx k$, and the rank of a projective module $P$ is by definition the image of a representative of $P$ in ${\bf Z}$.
What I can't see is how this homomorphism is defined, and why $K_0 k \approx k$. Can anyone spell this out please?