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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$K$-theory and right ideal class groups for HNP rings
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by Timothy J. Hodges PDF
Trans. Amer. Math. Soc. 302 (1987), 751-767 Request permission


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 ${K_0}$.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 302 (1987), 751-767
  • MSC: Primary 16A14; Secondary 16A33, 16A54, 18F25, 19A49
  • DOI:
  • MathSciNet review: 891645