A Generalization of Posner’s Theorem on Derivations in Rings

Fuad Ali Ahmed Almahdi (), Abdellah Mamouni () and Mohammed Tamekkante ()
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Fuad Ali Ahmed Almahdi: King Khalid University
Abdellah Mamouni: University Moulay Ismail
Mohammed Tamekkante: University Moulay Ismail

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 1, 187-194

Abstract: 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.

Keywords: Prime and semiprime rings; Posner’s result; 16N60; 16W10; 16W25 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s13226-020-0394-8

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