Abstract
In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an arbitrary ring, P be a prime ideal of R, and d be a derivation of R. If [[x, d(x)], y] ∈ P for all x, y ∈ R, then d(R) ⊆ P or R/P is commutative. In particular, if R is semiprime and d is a centralizing derivation of R, we prove that either R is commutative or there exists a minimal prime ideal P of R such that d(R) ⊆ P. As a consequence, we show that for any semiprime ring with a centralizing derivation there exists at least a minimal prime ideal P such that d(P) ⊆ P.
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Acknowledgment
The authors would like to express their gratitude to King Khalid University, Saudi Arabia for providing administrative and technical support. The authors would like also to thank the anonymous referee for his/her comments and suggestions, which have substantially improved the paper.
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Almahdi, F.A.A., Mamouni, A. & Tamekkante, M. A Generalization of Posner’s Theorem on Derivations in Rings. Indian J Pure Appl Math 51, 187–194 (2020). https://doi.org/10.1007/s13226-020-0394-8
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DOI: https://doi.org/10.1007/s13226-020-0394-8