Let R be a semiprime ring with center Z ( R ) and θ be a surjective homomorphism. In this paper, we prove that T is a θ -centralizer if one of the following holds: (i) T ( x ) θ ( y ) = θ ( x ) T ( y ) for all x, y ∈ R , where T is a mapping. (ii) T ( x ), θ ( x ) = 0 for all x ∈ R , where T is a left θ -centralizer. (iii) 2 T ( xyx ) = T ( x ) θ ( y ) θ ( x ) + θ ( x ) θ ( y ) T ( x ) for all x , y ∈ R , where T is an additive mapping, θ ( Z ( R )) = Z ( R ) and R is 2-torsion free.
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