带有恩格尔条件的广义导子

Generalized Derivations with Engel Conditions

令R是非交换的素环,I是环R的非零右理想,g是R的广义导子,满足[g(rk),rk]n=0,r∈I,k,n是固定的正整数,则存在c∈U,U是环R的右Utumi商环,对适当的α∈C,满足g(x)=cx,且(c-α)I=0,特别地,有g(x)=xα,x∈I.

Let R be a noncommutative prime ring,I be a nonzero right ideal of R,and g be a generalized derivation of R,such that[g（ r^k）,r^k]n= 0 for all r∈I,where k and n are fixed positive integers. Then there exists c∈U,U is the right Utumi quotient ring of R,such that g（ x） = cx and（ c- α） I = 0 for a suitable α∈C. In particular we have that g（ x） = xα,x∈I.