## Generalized coderivations of bicomodules.(English)Zbl 1346.16029

In a previous paper, [Int. J. Pure Appl. Math. 77, No. 4, 579-593 (2012; Zbl 1253.16041)], the author extended a generalized derivation to a map from a bimodule to a bimodule. By dualizing, it is possible to extend the definition of a generalized coderivation to a map from a bicomodule to a bicomodule over corings. The main goal of this paper is to investigate this new generalized coderivation. In particular, in section $$3$$ the author constructs a universal generalized coderivation. He shows in section $$4$$ that, for each $$(\mathcal{D,C})$$-bicomodule $$N$$ over corings $$\mathcal C$$ and $$\mathcal D$$, there exists a $$(\mathcal{D,C})$$-bicomodule $$\mathcal U(N)$$ that is isomorphic to the cotensor product of $$N$$ and $$\mathcal U(\mathcal D\otimes_R\mathcal C)$$. The paper finishes giving in section $$5$$ a characterization of a coseparable coring.

### MSC:

 16T15 Coalgebras and comodules; corings 16W25 Derivations, actions of Lie algebras

Zbl 1253.16041
Full Text:

### References:

 [1] T. Brzeziński, The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebr. Represent. Theory, 5 (2002), 389-410. · Zbl 1025.16017 [2] T. Brzeziński and R. Wisbauer, Corings and Comodules, Cambridge University Press, Cambridge, 2003. · Zbl 1035.16030 [3] Y. Doi, Homological coalgebra, J. Math. Soc. Japan, 33 (1981), 31-50. · Zbl 0459.16007 [4] J. Gómez-Torrecillas and A. Louly, Coseparable corings, Comm. Algebra, 31 (2003), 4455-4471. · Zbl 1037.16027 [5] F. Guzman, Cointegrations, relative cohomology for comodules, and coseparable corings, J. Algebra, 126 (1989), 211-224. · Zbl 0684.16018 [6] H. Komatsu, Generalized derivations of bimodules, Int. J. Pure Appl. Math., 77 (2012), 579-593. · Zbl 1253.16041 [7] R. G. Larson, Coseparable Hopf algebras, J. Pure Appl. Algebra, 3 (1973), 261-267. · Zbl 0276.18014 [8] A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22 (1980), 145-149. · Zbl 0449.16004 [9] A. Nakajima, On categorical properties of generalized derivations, Sci. Math., 2 (1999), 345-352. · Zbl 0968.16018 [10] A. Nakajima, On generalized coderivations, Int. Electron. J. Algebra, 12 (2012), 37-52. · Zbl 1254.16026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.