Abstract
For a ring R, let ▪ denote its reduced Grothendieck group. The purpose of this note is to establish (i) every f.g. torsion, or torsion-free, Abelian group is ▪ for some local Noetherian domain R, (ii) for every finitely generated Abelian group G, there exists a local Noetherian domain R for which ▪ is a finitely generated group H that contains a direct summand isomorphic to G. These are partial answers to the problem of Kaplansky: Given an Abelian group G, is there a local Noetherian domain R with ▪ ≈ G ([2, p. 68]).
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