# A generalization of Posner's theorem on generalized derivations in rings

Document Type : Research Article

Authors

1 Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India.

2 Department of Mathematics, Sivas Cumhuriyet University, Faculty of Science, Sivas, Turkey.

Abstract

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.

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