Abstract: Let R be a prime ring with center Z and extended centroid C, We have proven the following results. (1) Let f(x) and g(x) be non-zero derivations in prime ring R, supposing that there exists a,b∈R such that af(x)+bg(x) is a derivation of R and commuted on it, then f(x)=λx+ζ(x), g(x)=λ′x+ζ′(x), λ,λ′∈C, additive map ζ,ζ′: R→C; (2) Let R be a ring, a biadditive map G: R×R→R is the symmetric bi-derivation of R, if ［G(x,x),x］∈Z, char R≠2, then R is commuting; (3) let R be a prime ring of char R≠2 and d ₁,d₂ be non-zero derivations in R, if d₁d₂ (R)∈Z, then R is commuting.

Key words: prime ring, maximal right ring of quotient, derivation, extended centriod