A perturbation of ring derivations on Banach algebras

Suppose A is a Banach algebra and suppose f :A→A is an approximate ring derivation in the sense of Hyers–Ulam–Rassias. This stability phenomenon was introduced for the first time in the subject of functional equations by Th.M. Rassias [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer.Math. Soc. 72 (1978) 297–300]. If A has an approximate identity, or if A is semisimple and commutative, then we prove that f is an exact ring derivation. © 2005 Elsevier Inc. All rights reserved.