ISHIKAWA Hirofumi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (00108101)
HIRANO Yasuyuki Okayama University, Faculty of Science, Professor, 理学部, 助教授 (90144732)
IKEHATA Shuichi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (20116429)
KAJIWARA Tsuyoshi Okayama University, Faculty of Environmental Science and Technology, Associate Professor, 環境理工学部, 助教授 (50169447)
1. The notion of P-Galois extensions was introduced by K. Kishimoto. This notion contains that the usual Galois extensions, purely inseparable extensions and Hopf Galois extensions. We determine all cubic P-Galois extensions over a field except that P is a cyclic group. It is not known that the isomorphism classes of P-Galois extensions has a group structure or not. In our case, the isomorphism classes does not work well.
2. Let a be a ring and an M-bimodule. Let f be an additive map from A to M and ω an element in M. (f, ω) is called a generalized derivation of = f(x)y+xf(y)+xωy (x, y ∈ A). (f, ω) is a Bresar's generalized derivation and if A has an identity, they are equal. We give elementary relations of derivations, generalized derivations and Bresar's one and determine a functional relation between gDer(A, M), the set of all generalized derivations and Der(A, M), the set of all derivations from A to M. Using this result, we give a split short exact sequence with respect to M, gDer(A, M) and Der(A, M). Moreover, the universal mapping property for generalized derivations is given. The notion of generalized derivation is extended to Jordan and Lie derivations and we can get similar results to generalized Jordan derivations. Generalized higher derivations are also discussed.
In the stand point of Hopf algebras, the action of generalized derivation is a comodule algebra action. Therefore we have an application of these maps to Hopf Galois theory.