Abstract
Let \(f:R\rightarrow R\) be an additive map** and g be a function of R. If \(f\left( xy \right) {=}f\left( x \right) \) \(y{+}g\left( x \right) f\left( y \right) {=}f\left( x \right) g\left( y \right) {+}xf\left( y \right) \) and \(fg\left( x \right) {=}gf\left( x \right) \) for all \(x,y{\in }R\) then f is called a semi-derivation associated with g. Let R be a prime ring with characteristic different from two and \(\lambda , \mu {,\, }\sigma {,\, }\tau \) automorphisms of R. Let b be a nonzero element of R and \(I,J {,\, }U\) be nonzero ideals of R such that \(g{(}I{)\ne 0}\). If one of the following conditions holds then R is commutative: \(f{(}I{)\subset }C_{\lambda {,}\mu }\left( J \right) , \quad bf\left( R \right) {\subset }C_{\lambda {,}\mu }\left( R \right) , \quad {\, }f\left[ I{,}J \right] _{\lambda {,}\mu }{=0}, \quad \left[ f\left( I \right) {,}J \right] _{\sigma {,\, }\tau }{\subset }C_{\lambda {,}\mu })\) \(\left( U \right) , \quad \left[ f\left( x \right) {,}x \right] _{{1,}\mu }{=0}\) or \(\left[ f\left( x \right) {,}g{(}x{)} \right] _{\lambda {,1}}{=0}\), for all \(\, x\epsilon I,\, \left[ f\left( I \right) {,}f\left( R \right) \right] _{\lambda {,}\mu }{=0}\). Moreover, we proved that \(\left[ f\left( I \right) {,}a \right] _{\sigma {,\, }\tau }{=0}\) if and only if \({f\left[ I{,}a \right] }_{\sigma {,\, }\tau }{=0}\).
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08 July 2024
An Erratum to this paper has been published: https://doi.org/10.1007/s13226-024-00665-6
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The author would like to thank the referees for their useful comments and suggestions. This work has been supported by the Kocaeli University Scientific Research Projects Coordination Unit (ID:1599).
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Communicated by B. Sury.
“The original online version of this article was revised: ” Because of typesetting mistake, (xy) was incorrectly written as (x,y) for three times. The original article has been corrected.
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Güven, E. Commutativity of semi-derivative prime rings. Indian J Pure Appl Math (2023). https://doi.org/10.1007/s13226-023-00484-1
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DOI: https://doi.org/10.1007/s13226-023-00484-1