Generalized Derivations of Prime Rings

Huang Shuliang

International Journal of Mathematics and Mathematical Sciences, 2007, vol. 2007, 1-6

Abstract:

Let R be an associative prime ring, U a Lie ideal such that u 2 ∈ U for all u ∈ U . An additive function F : R → R is called a generalized derivation if there exists a derivation d : R → R such that F ( x y ) = F ( x ) y + x d ( y ) holds for all x , y ∈ R . In this paper, we prove that d = 0 or U ⊆ Z ( R ) if any one of the following conditions holds: (1) d ( x ) ∘ F ( y ) = 0 , (2) [ d ( x ) , F ( y ) = 0 ] , (3) either d ( x ) ∘ F ( y ) = x ∘ y or d ( x ) ∘ F ( y ) + x ∘ y = 0 , (4) either d ( x ) ∘ F ( y ) = [ x , y ] or d ( x ) ∘ F ( y ) + [ x , y ] = 0 , (5) either d ( x ) ∘ F ( y ) − x y ∈ Z ( R ) or d ( x ) ∘ F ( y ) + x y ∈ Z ( R ) , (6) either [ d ( x ) , F ( y ) ] = [ x , y ] or [ d ( x ) , F ( y ) ] + [ x , y ] = 0 , (7) either [ d ( x ) , F ( y ) ] = x ∘ y or [ d ( x ) , F ( y ) ] + x ∘ y = 0 for all x , y ∈ U .

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:085612

DOI: 10.1155/2007/85612

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