ON GENERALIZED CODERIVATIONS

Authors : Atsushi NAKAJİMA

Pages : 37-52

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Publication Date : 2012-12-01

Article Type : Research Paper

Abstract :Let C insert ignore into journalissuearticles values(resp. A); be a coalgebra insert ignore into journalissuearticles values(resp. algebra); over a commutative ring R and M insert ignore into journalissuearticles values(resp. N); a C-bicomodule insert ignore into journalissuearticles values(resp. an A-bimodule);. We define a dual notion of a generalized derivation from A to N in the sense of the paper On categorical properties of generalized derivations, Sci. Math., 2insert ignore into journalissuearticles values(3); insert ignore into journalissuearticles values(1999);, 345-352, by A. Nakajima, which we call a generalized coderivation from M to C. We give some elementary properties of generalized coderivations and discuss the relations of the set of generalized coderivations gCoderinsert ignore into journalissuearticles values(M, C); between the set of generalized derivations gDerinsert ignore into journalissuearticles values(C∗, M∗); for their dual algebra C∗ and module M∗. Using these coderivations, we define a notion of a weakly coseparable coalgebra which is a dual notion of a weakly separable algebra defined in the paper of N. Hamaguchi and A. Nakajima, Weakly separable polynomials insert ignore into journalissuearticles values(in preparation);, and give related examples of coseparable coalgebras.
Keywords : derivation, generalized derivation, coderivation, generalized coderivation, weakly coseparable coalgebra