Generalized derivations on semiprime rings
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Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that (F([x,y]))^n=[x,y] for all x, y∈ I. Then either R is commutative or n=1, d=0 and F is the identity map on R. Moreover in case R is a semiprime ring and (F([x,y]))^n=[x, y] for all x, y∈ R, then either R is commutative or n=1, d(R)⊆ Z(R), R contains a non-zero central ideal and F(x)-x ∈ Z(R) for all x∈ R.
Abstract
Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that (F([x,y]))^n=[x,y] for all x, y∈ I. Then either R is commutative or n=1, d=0 and F is the identity map on R. Moreover in case R is a semiprime ring and (F([x,y]))^n=[x, y] for all x, y∈ R, then either R is commutative or n=1, d(R)⊆ Z(R), R contains a non-zero central ideal and F(x)-x ∈ Z(R) for all x∈ R.
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