| Abstrakt: |
Let R be a semiprime ring, U a square-closed Lie ideal of R and D : R × R → R a symmetric bi-derivation and d be the trace of D. In the present paper, we prove that the R contains a nonzero central ideal if any one of the following holds: i) d x y ± x g (y) ∈ Z , ii) [ d (x) , y ] = ± [ x , g (y) ] , iii) d (x) ∘ y = ± x ∘ g (y) , iv) [ d (x) , y ] = ± x ∘ g (y) , v) d ([ x , y ]) = [ d (x) , y ] + [ d (y) , x ] , vi) d (x y) ± x y ∈ Z , vii) d (x y) ± y x ∈ Z , viii) d (x y) ± [ x , y ] ∈ Z , ix) d (x y) ± x ∘ y ∈ Z , x) g (x y) + d (x) d (y) ± x y ∈ Z , xi) g (x y) + d (x) d (y) ± y x ∈ Z , xii) g ([ x , y ]) + [ d (x) , d (y) ] ± [ x , y ] ∈ Z , xiii) g (x ∘ y) + d (x) ∘ d (y) ± x ∘ y ∈ Z , for all x , y ∈ U , where G : R × R → R is symmetric bi-derivation such that g is the trace of G. [ABSTRACT FROM AUTHOR] |
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