Skip to main content
Log in

An additivity principle for Goldie rank

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

LetA be a noetherian ring. In generalA will not admit a classical Artinian ring of quotients. Yet a problem in enveloping algebras leads one to consider the possible embedding ofA in a prime ringB which is finitely generated as a left and a rightA module. Under certain additional technical assumptions, it is shown that the setS of regular elements ofA is regular inB and is an Ore set in bothA andB withS −1 A andS −1 B Artinian. This enables one to establish the following additivity principle for Goldie rank. Let {P 1,P 2, …P 1} be the set of minimal primes ofA. Then under the above conditions it is shown that there exist positive integersz 1,z 2, …,z, such that \(\sum\limits_{i = 1}^r {z_i rk} (A/Pi) = rk B,\), where rk denotes Goldie rank. This applies to the study of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Borho,Recent advances in the enveloping algebras of semi-simple Lie algebras, Sem. Bourbaki, No. 489, Nov. 1976.

  2. W. Borho and H. Kraft,Über die Gelfand-Kirillov-Dimension, Math. Ann.220 (1976), 1–24.

    Article  MATH  MathSciNet  Google Scholar 

  3. W. Borho, P. Gabriel and R. Rentschler,Primideale in Einhüllenden auflösbarer Liealgebren, LN 357, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

    Google Scholar 

  4. M. Duflo,Sur la clasification des idéaux primitifs dans l'algèbre envelopante d'une algèbre de Lie semi-simple, Ann. Math.105 (1977), 107–130.

    Article  MathSciNet  Google Scholar 

  5. N. Jacobson,Structure of Rings, Amer. Math. Soc. Colloq. Publ., Vol. XXXVII, 1974, Chap. III.

  6. A. Joseph,Towards the Jantzen conjecture, perprint, Orsay, Nov. 1977.

  7. A. Joseph,Towards the Jantzen conjecture II, preprint, Jerusalem, Feb. 1978.

  8. L. W. Small,Orders in Artinian rings, J. Algebra4 (1966), 13–41.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This paper was written while the authors were guests of the Institute for Advanced Studies, The Hebrew University of Jerusalem. The first author was on leave of absence from the Centre Nationale de la Recherche Scientifique, France.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Joseph, A., Small, L.W. An additivity principle for Goldie rank. Israel J. Math. 31, 105–114 (1978). https://doi.org/10.1007/BF02760541

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02760541

Keywords

Navigation