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
- M. Breˇsar, On the distance of the composition of two derivations to the gener- alized derivations, Glasgow Math. J., 33(1) (1991), 89-93.
- F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York 181 (1971).
- Y. Doi, Homological coalgebra, J. Math. Soc. Japan, 33 (1981), 31-50.
- F. Guzman, Cointegrations, Relative Cohomology for Comodules, and Cosep- arable Corings, J. Algebra, 126 (1989), 211-224.
- N. Hamaguchi and A. Nakajima, Weakly separable polynomials, in preparation.
- R. G. Larson, Coseparable Hopf algebras, J. Pure Appl. Algebra, 3 (1973), 261-267.
- A. Nakajima, Cosemisimple coalgebras and coseparable coalgebras over coalge- bras, Math. J. Okayama Univ., 21 (1979), 125-140.
- A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22 (1980), 145-149.
- A. Nakajima, On categorical properties of generalized derivations, Sci. Math., 2(3) (1999), 345-352.
- M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. Atsushi Nakajima
- 1, Nakaku, Nakai,
- Okayama 703-8205, Japan
- e-mail: a2017bj.naka@hi2.enjoy.ne.jp
Abstract
References
- M. Breˇsar, On the distance of the composition of two derivations to the gener- alized derivations, Glasgow Math. J., 33(1) (1991), 89-93.
- F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York 181 (1971).
- Y. Doi, Homological coalgebra, J. Math. Soc. Japan, 33 (1981), 31-50.
- F. Guzman, Cointegrations, Relative Cohomology for Comodules, and Cosep- arable Corings, J. Algebra, 126 (1989), 211-224.
- N. Hamaguchi and A. Nakajima, Weakly separable polynomials, in preparation.
- R. G. Larson, Coseparable Hopf algebras, J. Pure Appl. Algebra, 3 (1973), 261-267.
- A. Nakajima, Cosemisimple coalgebras and coseparable coalgebras over coalge- bras, Math. J. Okayama Univ., 21 (1979), 125-140.
- A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22 (1980), 145-149.
- A. Nakajima, On categorical properties of generalized derivations, Sci. Math., 2(3) (1999), 345-352.
- M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. Atsushi Nakajima
- 1, Nakaku, Nakai,
- Okayama 703-8205, Japan
- e-mail: a2017bj.naka@hi2.enjoy.ne.jp
Details
| Other ID | JA52RZ66AA |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | December 1, 2012 |
| Published in Issue | Year 2012 Volume: 12 Issue: 12 |