One can construct topological spaces with prescribed homotopy groups or, say, homology groups. 
But is it possible to construct a ring with any given $K\_0$ group? What about $K\_1$ group et.c.?


I know very little about K-theory, so this question might be silly.


Thanks!



ring with prescribed K group
Every abelian group $G$ is the class group of some Dedekind domain $R$ (theorem of Luther Claborn), so we have $K\_0^{red}(R)= G$. 


