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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 ${\cal R}$ into a Banach-algebra ${\cal B}$. The main result of the paper on Hyers-Ulam stability is: Let $f:{\cal R}\to{\cal B}$ and let $\varepsilon,\delta>0$. If $\|f(x+y)-f(x)-f(y) \|\le \varepsilon$ and $\|f(xy)- f(x)f(y)\|\le \delta$ for all $x,y\in {\cal R}$, then there is exactly one ring homomorphism $h:{\cal R}\to {\cal B}$ such that $\|f(x)-h(x) \|\le\varepsilon$ for all $x\in {\cal R}$. This extends Theorem 5 of {\it 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 ${\cal R}$ is a normed algebra.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges
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##### References:
 [1] Aczél, J.: Lectures on functional equations and their application. (1966) · Zbl 0139.09301 [2] Bourgin, D. G.: Approximately isometric and multiplicative transformations on continuous function rings. Duke math. J. 16, 385-397 (1949) · Zbl 0033.37702 [3] Gajda, Z.: On stability of additive mappings. Internat. J. Math. math. Sci. 14, 431-434 (1991) · Zbl 0739.39013 [4] Hyers, D. H.: On the stability of the linear functional equation. Proc. nat. Acad. sci. USA 27, 222-224 (1941) · Zbl 0061.26403 [5] Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables. (1998) · Zbl 0907.39025 [6] Johnson, B. E.: Approximately multiplicative maps between Banach algebras. J. London math. Soc. (2) 37, 294-316 (1988) · Zbl 0652.46031 [7] Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. amer. Math. soc. 72, 297-300 (1978) · Zbl 0398.47040 [8] Ulam, S. M.: A collection of mathematical problems. (1960) · Zbl 0086.24101