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# Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

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1572-9176
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# Identities related to generalized derivations in prime ∗-rings

Abdelkarim Boua
• Department of Mathematics, Physics and Computer Science, Polydisciplinary Faculty, Sidi Mohammed Ben Abdellah University of Fez, LSI, Taza, Morocco
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• De Gruyter OnlineGoogle Scholar
/ Mohammed Ashraf
Published Online: 2019-10-15 | DOI: https://doi.org/10.1515/gmj-2019-2056

## Abstract

Let $\mathcal{ℛ}$ be a prime ring with center $Z\left(\mathcal{ℛ}\right)$ and $*$ an involution of $\mathcal{ℛ}$. Suppose that $\mathcal{ℛ}$ admits generalized derivations F, G and H associated with a nonzero derivation f, g and h of $\mathcal{ℛ}$, respectively. In the present paper, we investigate the commutativity of a prime ring $\mathcal{ℛ}$ satisfying any of the following identities: (i) $\left[F\left(x\right),F\left({x}^{*}\right)\right]=0$, (ii) $\left[F\left(x\right),F\left({x}^{*}\right)\right]=±\left[x,{x}^{*}\right]$, (iii) $F\left(x\right)\circ F\left({x}^{*}\right)=0$, (iv) $F\left(x\right)\circ F\left({x}^{*}\right)=±\left(x\circ {x}^{*}\right)$, (v) $\left[F\left(x\right),{x}^{*}\right]±\left[x,G\left({x}^{*}\right)\right]=0$, (vi) $F\left(x{x}^{*}\right)\in Z\left(\mathcal{ℛ}\right)$, (vii) $F\left(x\right)G\left({x}^{*}\right)±H\left(x\right){x}^{*}\in Z\left(\mathcal{ℛ}\right)$, (viii) $F\left(\left[x,{x}^{*}\right]\right)±\left[x,{x}^{*}\right]\in Z\left(\mathcal{ℛ}\right)$, (ix) $F\left(x\circ {x}^{*}\right)±x\circ {x}^{*}\in Z\left(\mathcal{ℛ}\right)$, (x) $\left[F\left(x\right),{x}^{*}\right]±\left[x,G\left({x}^{*}\right)\right]\in Z\left(\mathcal{ℛ}\right)$, (xi) $F\left(x\right)\circ {x}^{*}±x\circ G\left({x}^{*}\right)\in Z\left(\mathcal{ℛ}\right)$ for all $x\in \mathcal{ℛ}$. Finally, the restrictions imposed on the hypotheses have been justified by an example.

MSC 2010: 16N60; 16U80; 16W25

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## About the article

Received: 2017-03-03

Accepted: 2017-10-26

Published Online: 2019-10-15

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,

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