素环上的导子

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, char R≠2, 则R是交换的; (3) 若R是char R≠2的素环, d₁,d₂是R上非零导子, 且d< sub>1d₂(R)∈Z, 则R是交换的.

Derivations in prime rings

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 d₁,d₂ be non-zero derivations in R, if d₁d₂ (R)∈Z, then R is commuting.