Generalized Derivations and Left Ideals in Prime and Semiprime Rings

الملخص EN

Let R be an associative ring, λ a nonzero left ideal of R, d:R→R a derivation and G:R→R a generalized derivation.

In this paper, we study the following situations in prime and semiprime rings: (1) G(x∘y)=a(xy±yx); (2) G[x,y]=a(xy±yx); (3) d(x)∘d(y)=a(xy±yx); for all x,y∈λ and a∈{0,1,-1}.