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Galois Theory with Infinitely Many Idempotents1)

Published online by Cambridge University Press:  22 January 2016

O.E. Villamayor  and
D. Zelinsky
O.E. Villamayor
Affiliation:
Northwestern University
D. Zelinsky
Affiliation:
Northwestern University
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In 1942 Artin proved the linear independence, over a field S, of distinct automorphism of S; in other words if G is a finite group of automorphisms of S and R is the fixed field, then Horn^S, S) is a free S-module with G as basis. Since then, this last condition (“S is G-Galois”) or its equivalents have been used as a postulate in all the Galois theories of rings that are not fields, for example by Dieudonné, Jacobson, Azumaya and Nakayama for noncommutative rings and then in [AG, Appendix] and [CUR] for commutative rings. When S has no idempotents but 0 and 1, [CHR] proves that the ordinary fundamental theorem of Galois theory holds with no real change from the classical, field case.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 35 , July 1969 , pp. 83 - 98
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

Footnotes

1)

This research was made possible by N.S.F. Grant GP-1649, Travel Grant GP-8231, and the Conference on Rings and Modules at Oberwolfach, March 1968.

References

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[CE] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, Princeton, 1956.Google Scholar
[CHR] Chase, S.U., Harrison, D.K., Rosenberg, A., Galois theory and Galois cohomology of commutative rings, Memoirs Amer. Math. Soc. No. 52, 1965.Google Scholar
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[VZ] Villamayor, O.E. and Zelinsky, D., Galois theory for rings with finitely many idempotents, Nagoya Math. J. 27 (1966) 721–731.CrossRefGoogle Scholar