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

HTML articles powered by AMS MathViewer

- by Timothy J. Hodges PDF
- Trans. Amer. Math. Soc.
**302**(1987), 751-767 Request permission

## 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 ${K_0}$.## References

- Hyman Bass,
*Algebraic $K$-theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0249491** - David Eisenbud and J. C. Robson,
*Modules over Dedekind prime rings*, J. Algebra**16**(1970), 67–85. MR**289559**, DOI 10.1016/0021-8693(70)90041-4 - David Eisenbud and J. C. Robson,
*Hereditary Noetherian prime rings*, J. Algebra**16**(1970), 86–104. MR**291222**, DOI 10.1016/0021-8693(70)90042-6 - K. R. Goodearl,
*Localization and splitting in hereditary noetherian prime rings*, Pacific J. Math.**53**(1974), 137–151. MR**354748**, DOI 10.2140/pjm.1974.53.137 - K. R. Goodearl,
*The state space of $K_{0}$ of a ring*, Ring theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978) Lecture Notes in Math., vol. 734, Springer, Berlin, 1979, pp. 91–117. MR**548125** - K. R. Goodearl and R. B. Warfield Jr.,
*Simple modules over hereditary Noetherian prime rings*, J. Algebra**57**(1979), no. 1, 82–100. MR**533102**, DOI 10.1016/0021-8693(79)90210-2 - K. R. Goodearl and R. B. Warfield Jr.,
*State spaces of $K_{0}$ of Noetherian rings*, J. Algebra**71**(1981), no. 2, 322–378. MR**630603**, DOI 10.1016/0021-8693(81)90181-2 - Robert Gordon and J. C. Robson,
*Krull dimension*, Memoirs of the American Mathematical Society, No. 133, American Mathematical Society, Providence, R.I., 1973. MR**0352177** - H. Jacobinski,
*Two remarks about hereditary orders*, Proc. Amer. Math. Soc.**28**(1971), 1–8. MR**272807**, DOI 10.1090/S0002-9939-1971-0272807-7
A. V. Jategaonkar, - James Kuzmanovich,
*Localizations of HNP rings*, Trans. Amer. Math. Soc.**173**(1972), 137–157. MR**311699**, DOI 10.1090/S0002-9947-1972-0311699-X - John Milnor,
*Introduction to algebraic $K$-theory*, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR**0349811** - Daniel Quillen,
*Higher algebraic $K$-theory. I*, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR**0338129** - I. Reiner,
*Maximal orders*, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR**0393100** - J. C. Robson,
*Idealizers and hereditary Noetherian prime rings*, J. Algebra**22**(1972), 45–81. MR**299639**, DOI 10.1016/0021-8693(72)90104-4 - J. T. Stafford,
*Generating modules efficiently: algebraic $K$-theory for noncommutative Noetherian rings*, J. Algebra**69**(1981), no. 2, 312–346. MR**617082**, DOI 10.1016/0021-8693(81)90208-8 - J. T. Stafford and R. B. Warfield Jr.,
*Hereditary orders with infinitely many idempotent ideals*, J. Pure Appl. Algebra**31**(1984), no. 1-3, 217–225. MR**738216**, DOI 10.1016/0022-4049(84)90087-2 - J. T. Stafford and R. B. Warfield Jr.,
*Constructions of hereditary Noetherian rings and simple rings*, Proc. London Math. Soc. (3)**51**(1985), no. 1, 1–20. MR**788847**, DOI 10.1112/plms/s3-51.1.1 - Bo Stenström,
*Rings of quotients*, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR**0389953**, DOI 10.1007/978-3-642-66066-5 - R. B. Warfield Jr.,
*The number of generators of a module over a fully bounded ring*, J. Algebra**66**(1980), no. 2, 425–447. MR**593603**, DOI 10.1016/0021-8693(80)90096-4 - Abraham Zaks,
*Hereditary Noetherian rings*, J. Algebra**29**(1974), 513–527. MR**349750**, DOI 10.1016/0021-8693(74)90087-8

*Localisation in Noetherian rings*, London Math. Soc. Lecture Note Series 89, Cambridge Univ. Press, Cambridge, 1985.

## 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: https://doi.org/10.1090/S0002-9947-1987-0891645-1
- MathSciNet review: 891645