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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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##### References:
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