Abstract
Let \({\mathcal{A}}\) be a \({\mathbb{C}}\) -algebra, δ be a derivation on \({\mathcal{A}}\) and \({\mathcal{M}}\) be a left \({\mathcal{A}}\) -module. A linear map \({\tau : \mathcal{M} \rightarrow \mathcal{M}}\) is called a generalized derivation relative to δ if \({\tau(am)=a\tau(m)+\delta(a)m\,(a \in \mathcal{A}, m \in \mathcal{M})}\). In this article first we study the existence of generalized derivations. In particular we show that free modules and projective modules always have nontrivial generalized derivations relative to nonzero derivations of \({\mathcal{A}}\). Then we investigate the invariance of prime submodules under generalized derivations. Specifically we show that every minimal prime submodule of \({\mathcal{M}}\) is invariant under every generalized derivation. Moreover we obtain analogs of Posner’s theorem for generalized derivations. In the case that \({\mathcal{A}}\) is a Banach algebra and \({\mathcal{M}}\) is a Banach left \({\mathcal{A}}\) -module, we study the existence of continuous generalized derivations and automatic continuity of generalized derivations.
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References
Abbaspour G., Moslehian M.S., Niknam A.: Generalized derivations on modules. Bull. Iran. Math. Soc. 32(1), 21–32 (2006)
Abdelali Z.: On \({\phi}\) -derivations in Banach algebras. Commun. Alg. 34, 2437–2452 (2006)
Brešar M.: On the distance of compositions of two derivations of the generalized derivations. Glasgow Math. J. 33, 89–93 (1991)
Brešar M.: Derivations of noncommutative Banach algebras II. Arch. Math. 63, 56–59 (1994)
Brešar M., Mathieu M.: Derivations mapping into the radical, III. J. Funct. Anal. 133(1), 21–29 (1995)
Brešar M., Villena A.: The non-commutative Singer–Wermer conjecture and \({\phi}\) -derivations. J. Lond. Math. Soc. 66(2–3), 710–720 (2002)
Creedon T.: Derivations and prime ideals. Proc. Royal Irish Acad. 98(A2), 223–225 (1998)
Dales, H.G.: Banach algebras and automatic continuity. In: London Math. Soc. Monographs. Oxford Univ. Press, Oxford (2000)
Dauns J.: Prime modules. J. Reine Angew Math. 298, 156–181 (1978)
Ghahramani F., Runde V., Willis G.: Derivations on group algebras. Proc. Lond. Math. Soc. 80(3), 360–390 (2000)
Hvala B.: Generalized derivations in prime rings. Commun. Alg. 26(4), 1147–1166 (1998)
Lee T.K., Liu C.K.: Spectrally bounded \({\phi}\) -derivations on Banach algebras. Proc. Am. Math. Soc. 133(5), 1427–1435 (1900)
Lu C.P.: Prime submodules of modules. Comm. Math. Univ. Sancti Pauli 33, 61–69 (1984)
Mathieu M., Murphy G.: Derivations mapping into the radical. Arch. Math. 57, 469–474 (1991)
Mc Casland R.L., Smith P.F.: Prime submodules of Noetherian modules. Rocky Mt. J. Math. 23, 1041–1062 (1993)
Mirzavaziri M., Moslehian M.S.: σ-derivations in Banach algebras. Bull. Iran. Math. Soc. 32(1), 65–78 (2006)
Mirzavaziri M., Moslehian M.S.: Automatic continuity of σ-derivations in C*-algebras. Proc. Am. Math. Soc. 134(11), 3319–3327 (2006)
Posner E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8, 1093–1100 (1957)
Sinclair A.M.: Continuous derivations on Banach algebras. Proc. Am. Math. Soc. 20, 166–170 (1969)
Thomas M.P.: The image of a derivation is contained in the radical. Ann. Math. 128, 435–460 (1988)
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Esslamzadeh, G.H., Ghahramani, H. Existence, automatic continuity and invariant submodules of generalized derivations on modules. Aequat. Math. 84, 185–200 (2012). https://doi.org/10.1007/s00010-012-0127-0
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DOI: https://doi.org/10.1007/s00010-012-0127-0