Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 3 1 2022 01 01 A generalization of Posner's theorem on generalized derivations in rings 1 9 160787 10.30504/jims.2022.335190.1059 EN N. U. Rehman Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India. E. K. Sögütcü Department of Mathematics, Sivas Cumhuriyet University, Faculty of Science, Sivas, Turkey. 0000-0002-8328-4293 H. M. Alnoghashi Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India. 0000-0003-0253-6573 Journal Article 2022 03 25  In this paper, we generalize the Posner's theorem on generalized derivations in rings as follows: Let $\mathscr{A}$ be an arbitrary ring, $\mathscr{I}$ a non-zero ideal, $\mathscr{T}$ is a prime ideal of $\mathscr{A}$ such that $\mathscr{T}\subset \mathscr{I},$ and $\psi $ be a non-zero generalized derivation associated with a non-zero derivation $\rho $ of $\mathscr{A}.$ If one of the following conditions is satisfied: (i) $[\psi (x),x]\in \mathscr{T},$ (ii) $[[\psi (x),x],y]\in \mathscr{T},$ (iii) $\overline{[\psi (x),x]}\in \mathscr{Z}(\mathscr{A}/\mathscr{T})$ and (iv) $\overline{[[\psi (x),x],y]}\in \mathscr{Z}(\mathscr{A}/\mathscr{T})$ $\forall $ $x,y\in \mathscr{I},$ then $\rho (\mathscr{A})\subseteq \mathscr{T}$ or $\mathscr{A}/mathscr{T}$ is commutative. At the example, it is given that the hypothesis of the  theorems are necessary. https://jims.ims.ir/article_160787_83e824ba9c6c8ed98d6a76610333af8e.pdf