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On approximate ring homomorphisms. (English) Zbl 1014.39020
The subject is stability of Hyers-Ulam type and of Rassias type of ring homomorphisms from a ring \({\mathcal R}\) into a Banach-algebra \({\mathcal B}\).
The main result of the paper on Hyers-Ulam stability is: Let \(f:{\mathcal R}\to{\mathcal B}\) and let \(\varepsilon,\delta>0\). If \(\|f(x+y)-f(x)-f(y) \|\leq \varepsilon\) and \(\|f(xy)- f(x)f(y)\|\leq \delta\) for all \(x,y\in {\mathcal R}\), then there is exactly one ring homomorphism \(h:{\mathcal R}\to {\mathcal B}\) such that \(\|f(x)-h(x) \|\leq\varepsilon\) for all \(x\in {\mathcal R}\). This extends Theorem 5 of D. G. Bourgin’s paper [Duke Math. J. 16, 385–397 (1949; Zbl 0033.37702)].
The author modifies his proof to obtain a similar result about stability of Rassias type of ring homomorphisms in case \({\mathcal R}\) is a normed algebra.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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