On a theorem from Skew field constructions

Cohn, P. M.

doi:10.1090/S0002-9939-1982-0633265-0

On a theorem from Skew field constructions
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Proc. Amer. Math. Soc. 84 (1982), 1-7 Request permission

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

P. M. Cohn, Free rings and their relations, London Mathematical Society Monographs, No. 2, Academic Press, London-New York, 1971. MR 0371938
Paul Moritz Cohn, Algebra. Vol. 2, John Wiley & Sons, London-New York-Sydney, 1977. With errata to Vol. I. MR 0530404
Paul Moritz Cohn, Skew field constructions, London Mathematical Society Lecture Note Series, No. 27, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0463237

The universal field of fractions of a semifir

P. M. Cohn and Warren Dicks, On central extensions of skew fields, J. Algebra 63 (1980), no. 1, 143–151. MR 568568, DOI 10.1016/0021-8693(80)90029-0