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On a theorem from Skew field constructions

Author: P. M. Cohn
Journal: Proc. Amer. Math. Soc. 84 (1982), 1-7
MSC: Primary 16A39; Secondary 16A06
MathSciNet review: 633265
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Abstract: Let $ F$ be a skew field and $ C$ a central subfield, then the free $ F$-field on $ X$ centralizing $ C$ is denoted by $ {F_C}(X)$. The object is to prove the following theorem. Let $ F$ be a skew field with a central subfield $ C$, let $ E$ be a subfield of $ F$ and put $ k = E \cap C$; then there is a natural embedding of $ {E_k}(X)$ in $ {F_C}(X)$ if and only if $ E$ and $ C$ are linearly disjoint over $ k$.

This result replaces the erroneous Theorem 6.3.6 on p. 148 of the author's Skew field constructions, a counterexample to the latter (due to G. M. Bergman) is also described. The paper also includes an improved form of the specialization lemma (1.c.).

References [Enhancements On Off] (What's this?)

  • [1] P. M. Cohn, Free rings and their relations, London Math. Soc. Monographs, No. 2, Academic Press, London and New York, 1971. MR 0371938 (51:8155)
  • [2] -, Algebra, vol. 2, Wiley, Chichester, 1977. MR 0530404 (58:26625)
  • [3] -, Skew field constructions, London Math. Soc. Lecture Notes, No. 27, Cambridge Univ. Press, Cambridge, 1977. MR 0463237 (57:3190)
  • [4] -, The universal field of fractions of a semifir. III (to appear).
  • [5] P. M. Cohn and W. Dicks, On central extensions of skew fields, J. Algebra 63 (1980), 143-151. MR 568568 (82m:16014)

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Keywords: Skew field (division ring), centre, linearly disjoint, full, honest, semifir, fir, Ore domain, quaternions, specialization lemma
Article copyright: © Copyright 1982 American Mathematical Society