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On approximately higher ring derivations. (English) Zbl 1144.39024
Let ${\mathcal A}$ be an algebra and $n_0 \in \{0, 1, \dots, \}\cup \{\infty\}$. A sequence $(d_j)_{j=1}^{n_0}$ of mappings on ${\mathcal A}$ is called a higher ring derivation of rank $n_0$ if for each $0 \leq j \leq n_0$, $$d_j(ab) = \sum_{\ell=0}^j d_\ell(a)d_{j-\ell}(b) \qquad (a, b \in {\mathcal A}).$$ It is obvious that $d_0$ is a homomorphism and $d_1$ is a $d_0$-derivation in the sense of {\it M. Mirzavaziri} and {\it M. S. Moslehian} [Proc. Am. Math. Soc. 134, No. 11, 3319--3327 (2006; Zbl 1116.46061)]. In this paper the authors use ideas of {\it R. Badora} [J. Math. Anal. Appl. 276, No. 2, 589--597 (2002; Zbl 1014.39020)] and {\it T. Miura, G. Hirasawa} and {\it S.-E. Takahasi} [J. Math. Anal. Appl. 319, No. 2, 522--530 (2006; Zbl 1104.39025)] to establish the stability of higher derivations on Banach algebras as well as superstability of such mappings under the surjectivity of $d_0$.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
47B47Commutators, derivations, elementary operators, etc.
Full Text: DOI
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