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Polynomials over division rings

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Abstract

LetD be a division ring with a centerC, andD[X 1, …,X N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X 1, …,X m]-modules are finite-dimensionalD-spaces.

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References

  1. S. A. Amitsur,Algebras over infinite fields, Proc. Amer. Soc. 7 (1956), 35–48.

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Authors and Affiliations

  1. The Hebrew University of Jerusalem, Jerusalem, Israel

    S. A. Amitsur & Lance W. Small

  2. University of California San Diego, 92093, La Jolla, Calif., USA

    S. A. Amitsur & Lance W. Small

Authors
  1. S. A. Amitsur
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  2. Lance W. Small
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Additional information

This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.

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Amitsur, S.A., Small, L.W. Polynomials over division rings. Israel J. Math. 31, 353–358 (1978). https://doi.org/10.1007/BF02761500

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  • DOI: https://doi.org/10.1007/BF02761500

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