ON GENERALIZED CODERIVATIONS

Abstract

Let C (resp. A) be a coalgebra (resp. algebra) over a commutative ring R and M (resp. N) a C-bicomodule (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., 2(3) (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 gCoder(M, C) between the set of generalized derivations gDer(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 (in preparation), and give related examples of coseparable coalgebras.

Keywords

• derivation, generalized derivation, coderivation, generalized coderivation, weakly coseparable coalgebra

References

1. M. Breˇsar, On the distance of the composition of two derivations to the gener- alized derivations, Glasgow Math. J., 33(1) (1991), 89-93.
2. F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York 181 (1971).
3. Y. Doi, Homological coalgebra, J. Math. Soc. Japan, 33 (1981), 31-50.
4. F. Guzman, Cointegrations, Relative Cohomology for Comodules, and Cosep- arable Corings, J. Algebra, 126 (1989), 211-224.
5. N. Hamaguchi and A. Nakajima, Weakly separable polynomials, in preparation.
6. R. G. Larson, Coseparable Hopf algebras, J. Pure Appl. Algebra, 3 (1973), 261-267.
7. A. Nakajima, Cosemisimple coalgebras and coseparable coalgebras over coalge- bras, Math. J. Okayama Univ., 21 (1979), 125-140.
8. A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22 (1980), 145-149.

9. A. Nakajima, On categorical properties of generalized derivations, Sci. Math., 2(3) (1999), 345-352.
10. M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. Atsushi Nakajima
11. 1, Nakaku, Nakai,
12. Okayama 703-8205, Japan
13. e-mail: [e-mail]