Approximate Orthogonally Higher Ring Derivations

In this paper‎, ‎we prove that every orthogonally higher ring derivation is a higher ring derivation‎. ‎Also we find the general solution of the pexider orthogonally higher ring derivations‎‎\begin{align*}‎‎\left\{‎‎\begin{array}{lr}‎‎f_n(x+y)=g_n(x)+h_n(y)‎, ‎\;\left\langle x,y \right\rangle =۰,\\‎‎f_n(xy) = \sum_{i+j=n} g_i(x)h_j(y)‎.‎\end{array}‎‎\right‎.‎\end{align*}‎‎Then we prove that for any approximate pexider orthogonally higher ring derivation under some control functions \varphi(x,y) and \psi(x,y) ‎, ‎there exists a unique higher ring derivation D=\{d_n\}_{n=۰}^\infty ‎, ‎near \{f_n\}_{n=۰}^\infty ‎, ‎ \{g_n\}_{n=۰}^\infty and \{h_n\}_{n=۰}^\infty estimated by \varphi and \psi .

Keywords:

Approximation‎ , ‎Control function‎ , ‎Estimation‎ , ‎Higher derivation