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For a ring $R$ there are several groups that can be derived from it. The very first example is of course the unit group $R^*$.Another group would be its ideal class group/its picard group $Pic(R)$.These groups are just special cases/are contained of/in certain $K$-Groups of $R$.

Is it possibile to define K-groups up to a fixed number $n$,so that one is able to reconstruct each ring ,if we know its groups?

If this was the case,it would be very impressive result and im sure i wouldve heared about it. So i will weaken my condition to whatever kind of non trivial example you can offer.

Id say that the integer rings of cyclotomic fields are pretty close to what im looking for. If one allows only $p-$roots of unity ,the order of the unit group , is already enough to distinguisch between these rings. Thats the reason why i came up with this question.

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I think homotopy invariance $K'_i(A[T]) = K'_i(A)$ for $A$ Noetherian destroys your hopes.

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  • $\begingroup$ For A regular this hold for K groups, not to mention the K theory does not distinguish Morita equipment rings! $\endgroup$
    – user1437
    Commented Nov 8, 2013 at 19:34
  • $\begingroup$ Ok these arent the best news. Are there any groups asscociated to a ring that do not arise in the context of K-Theory? $\endgroup$
    – Jason Pioneer
    Commented Nov 9, 2013 at 17:45

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