素环上的导子

Derivations in prime rings

设R是中心为Z、扩张形心为C的素环,证明了:(1)设f(x),g(x)为R上非零导子,若af(x)+bg(x)亦是R上导子,且在R上交换,则f(x)=λx+ζ(x),g(x)=λ′x+ζ′(x),其中λ,λ′∈C,ζ,ζ′:R→C加性映射;(2)设R是环,双加性映射G:R×R→R是R上对称双导子,若[G(x,x),x]∈Z,charR≠2,则R是交换的;(3)若R是charR≠2的素环,d1,d2是R上非零导子,且d1d2(R)∈Z,则R是交换的.

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 ∈Z, char R≠2, then R is commuting; (3) let R be a prime ring of char R≠2 and d_1,d_2 be non-zero derivations in R, if (d_1d_2(R)∈Z), then R is commuting.