Payame Noor University (PNU) Control and Optimization in Applied Mathematics 2383-3130 7 1 2022 01 01 Approximate Orthogonally Higher Ring Derivations 93 106 9033 10.30473/coam.2021.58866.1161 EN Sayed Kahlil Ekrami Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran Journal Article 2021 05 01 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<br />‎\begin{align*}‎<br />‎\left\{‎<br />‎\begin{array}{lr}‎<br />‎f_n(x+y)=g_n(x)+h_n(y)‎, ‎\;\left\langle x,y \right\rangle =0,\\‎<br />‎f_n(xy) = \sum_{i+j=n} g_i(x)h_j(y)‎.<br />‎\end{array}‎<br />‎\right‎.<br />‎\end{align*}‎<br />‎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=0}^\infty $‎, ‎near $ \{f_n\}_{n=0}^\infty $‎, ‎$ \{g_n\}_{n=0}^\infty $ and $ \{h_n\}_{n=0}^\infty $ estimated by $ \varphi $ and $ \psi $. https://mathco.journals.pnu.ac.ir/article_9033_5416c7b4613626683072d096c1139c08.pdf