On modular extensions

Sato, Shizuka

doi:10.1090/S0002-9939-1990-1019282-4

On modular extensions
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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

Élément de Géométrie Algébique

16.8

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

Lectures in abstract algebra

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

Commutative rings and derivations

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