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Zbl 1014.39020
Badora, Roman
On approximate ring homomorphisms. (English)
[J] J. Math. Anal. Appl. 276, No.2, 589-597 (2002). ISSN 0022-247X

The subject is stability of Hyers-Ulam type and of Rassias type of ring homomorphisms from a ring ${\cal R}$ into a Banach-algebra ${\cal B}$.\par 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)].\par 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.
[Henrik Stetkaer (Aarhus)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: Hyers-Ulam stability; Rassias stability; Banach-algebra; ring homomorphisms

Citations: Zbl 0033.37702

Cited in: Zbl 1235.39030 Zbl 1196.39017 Zbl 1144.39024 Zbl 1109.39026

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