关于素环广义导子和对称双导的几点备注

摘要

Let R be a prime ring and I a nonzero ideal of R.If R admits a generalized derivation δ not the identity map,then R is commutative.Moreover,if R is a 2-torsion free prime ring and U a Lie ideal of R such that u2∈U for all u∈U and γ is a generalized derivation with d≠0,B:R×R→R is a symmetric biderivation associated with the trace function g(x)=B(x,x),then U(≌)Z(R) when one of the following conditions holds (1) γ acts as a homomorphism on U(2)2[x,y]-g(xy)+g(yx)∈Zfor all x,y∈U.%设R是素环,I是R的非零理想,如果R容许一个非单位映射的左乘子使得对所有x,Y∈I满足δ(x·y)=x·y或δ(x·y)+x·y=0,那么R可交换.此外,如果R是2-扭自由的素环,U是平方封闭的李理想.γ是伴随导子非零的广义导子,B:R×R→R是迹函数为g(x)=B(x,x)的对称双导,当下列条件之一成立时U为中心李理想(1)γ同态作用于U(2)2[x,y]-g(xy)+g(yx)∈Z(R)(3)2[x,y]+g(xy)-g(yx)∈Z(R)(4)2(x·y)=g(x)-g(y)(5)2(x·y)=g(y)-g(x)对所有的x,y∈U.