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On generalized left derivations in rings and Banach algebras

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Abstract

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let \({G: R \longrightarrow R}\) be a generalized Jordan left derivation with associated Jordan left derivation \({\delta: R \longrightarrow R}\). Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer (or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.

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  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Shakir Ali

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  1. Shakir Ali
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Ali, S. On generalized left derivations in rings and Banach algebras. Aequat. Math. 81, 209–226 (2011). https://doi.org/10.1007/s00010-011-0070-5

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