$K$-theory and right ideal class groups for HNP rings

Abstract:

Let $R$ be an hereditary Noetherian prime ring, let $S$ be a "Dedekind closure" of $R$ and let $\mathcal {T}$ be the category of finitely generated $S$-torsion $R$-modules. It is shown that for all $i \geq 0$, there is an exact sequence $0 \to {K_i}(\mathcal {T}) \to {K_i}(R) \to {K_i}(S) \to 0$. If $i = 0$, or $R$ has finitely many idempotent ideals then this sequence splits. A notion of "right ideal class group" is then introduced for hereditary Noetherian prime rings which generalizes the standard definition of class group for hereditary orders over Dedekind domains. It is shown that there is a decomposition ${K_0}(R) \cong {\text {Cl}}(R) \oplus F$ where $F$ is a free abelian group whose rank depends on the number of idempotent maximal ideals of $R$. Moreover there is a natural isomorphism ${\text {Cl}}(R) \cong {\text {Cl}}(S)$ and this decomposition corresponds closely to the splitting of the above exact sequence for ${K_0}$.