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On Conjugates in Division Rings

Published online by Cambridge University Press:  20 November 2018

Carl C. Faith
Carl C. Faith*
Affiliation:
The Pennsylvania State University
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Let D be a non-commutative division ring with centre C, and let Δ be a proper division subring not contained in C. In (4) Cartan raised the question: is it ever possible for each inner automorphism of D to induce an automorphism of Δ? As is well-known, Cartan (4, Théorème 4) with the aid of his Galois Theory answered this negatively in case D is a finite dimensional division algebra. Later Brauer (3), and Hua (8), using elegant, elementary methods, extended Cartan's theorem to arbitrary division rings.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 374 - 380
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Amitsur, S. A., Invariant submodules of simple rings, Proc. Amer. Math. Soc, 7 (1956), 987-989.Google Scholar
2. Baxter, W. E., Lie simplicity of a special class of associative rings, Proc. Amer. Math. Soc, 7 (1956), 855-863.Google Scholar
3. Brauer, R., On a theorem of H. Cartan, Bull. Amer. Math. Soc, 55 (1949), 619-620.Google Scholar
4. Cartan, H., Théorie de Galois pour les corps non commutatif, Ann. Ecole Norm. Sup., 64 (1947), 59-77.Google Scholar
5. Faith, C. C., Concerning non-invariant submodules of simple, matrix, and local matrix rings, Notices Amer. Math. Soc, 5 (1958), 372.Google Scholar
6. Hattori, A. and Toyoda, G., On the multiplicative group of simple algebras, J. Math. Soc Japan, 4 (1954), 262-265.Google Scholar
7. Herstein, I. N., Conjugates in division rings, Bull. Amer. Math. Soc, 7 (1956), 1021-1022.Google Scholar
8. Hua, L. K., Some properties of a sfield, Proc. Nat. Acad. Sci., 85 (1949), 533-537.Google Scholar
9. Jacobson, N., Structure theory for algebraic algebras of bounded degrees, Ann. Math., 46 (1945), 695-707.Google Scholar
10. Jacobson, N., Structure of rings, Amer. Math. Soc Colloquim Publications, 37 (New York, 1956).Google Scholar
11. Kasch, F., Eine Bemerkung ueber innere Automorphismen, Math. J. Okayama University, 6, (1957), 131-133.Google Scholar
12. Nagahara, T., On generating elements of Galois extensions of division rings, Math. J. Okayama Univ., 6 (1957), 182.Google Scholar
13. Nobusawa, N., On compact Galois groups of division rings, Osaka Math. J., 8 (1956), 43–50.Google Scholar
14. Rosenberg, A., The Cartan-Brauer-Hua theorem for matrix and local matrix rings, Proc Amer. Math. Soc, 7 (1956), 891-898.Google Scholar
15. Scott, W. R., On the multiplicative group of a division ring, Proc Amer. Math. Soc, 8 (1957), 305.Google Scholar
16. Tominaga, H., A note on conjugates, Math. J. Okayama Univ., 7 (1957), 75-76.Google Scholar
17. Wedderburn, J. H. M., A theorem on finite algebras, Trans. Amer. Math. Soc, 6 (1905), 349-352.Google Scholar