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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!

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up vote 3 down vote accepted

Every abelian group $G$ is the class group of some Dedekind domain $R$ (theorem of Luther Claborn), so we have $K_0^{red}(R)= G$.

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Very nice! What about getting prescribed non-reduced $K$-theory via non-commutative rings? –  Daniel Miller Apr 23 '13 at 15:33
    
Thanks a lot for your answer! Do you know any similar results about $K_1$? –  Sasha Patotski Apr 24 '13 at 13:01
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