A perturbation of ring derivations on Banach algebras

Suppose A is a Banach algebra and suppose 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.