On modular extensions
HTML articles powered by AMS MathViewer
- by Shizuka Sato PDF
- Proc. Amer. Math. Soc. 109 (1990), 621-626 Request permission
Abstract:
M. E. Sweedler has proved that modular extensions of fields are characterized by a tensor product of primitive elements, and also by the equivalent condition that the ground field is the fixed field under higher derivations. In this paper we shall give an extension of his work about modular field extensions to modular integral domain extensions. Moreover, we shall prove that a modular extension is an extension that the derivation algebra is generated by components of higher derivations under some conditions. For example, in a finite extension of a field, a modular extension is characterized by the fact that the derivation algebra is generated by components of higher derivations.References
-
A. Grothendieck, Élément de Géométrie Algébique IV, Étude Locale des Schémas et des Morphismes de Schémas, 16.8 (1967).
- Nickolas Heerema, Higher derivations and automorphisms of complete local rings, Bull. Amer. Math. Soc. 76 (1970), 1212–1225. MR 266916, DOI 10.1090/S0002-9904-1970-12609-X
- Robert G. Heyneman and Moss Eisenberg Sweedler, Affine Hopf algebras. I, J. Algebra 13 (1969), 192–241. MR 245570, DOI 10.1016/0021-8693(69)90071-4 N. Jacobson, Lectures in abstract algebra III, Van Nostrand Company, New York.
- H. F. Kreimer and N. Heerema, Modularity vs. separability for field extensions, Canadian J. Math. 27 (1975), no. 5, 1176–1182. MR 392951, DOI 10.4153/CJM-1975-123-2
- Yoshikazu Nakai, High order derivations. I, Osaka Math. J. 7 (1970), 1–27. MR 263804
- Yoshikazu Nakai, Kôtaro Kosaki, and Yasunori Ishibashi, High order derivations. II, J. Sci. Hiroshima Univ. Ser. A-I Math. 34 (1970), 17–27. MR 266905 Y. Nakai, Commutative rings and derivations (in Japanese), Kyoritsu shuppan.
- Yoshikazu Nakai, On a ring with a plenty of high order derivations, J. Math. Kyoto Univ. 13 (1973), 159–164. MR 314817, DOI 10.1215/kjm/1250523445
- Shizuka Satô, Notes on derivations of higher order, J. Sci. Hiroshima Univ. Ser. A-I Math. 33 (1969), 41–46. MR 248127, DOI 10.1080/00031305.1968.10480486
- Shizuka Satô, On purely inseparable algebras and P.H.D. rings, Trans. Amer. Math. Soc. 266 (1981), no. 2, 483–498. MR 617546, DOI 10.1090/S0002-9947-1981-0617546-6
- Moss Eisenberg Sweedler, Structure of inseparable extensions, Ann. of Math. (2) 87 (1968), 401–410. MR 223343, DOI 10.2307/1970711
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 621-626
- MSC: Primary 13B02; Secondary 12F05, 13B10, 13G05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1019282-4
- MathSciNet review: 1019282