Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds: 1. there exist α 2 C and a 2 U such that F(x) = [a, x] and G(x) =αx, for all x εR; 2. G is an usual derivation of R and there exists α 2 C such that F(x) = αx, for all x ε R; 3. there exist α, β 2 C and a derivation h of R such that F(x) =αx+h(x), G(x) = βx, for all x ε R, and αβ+h(β) = 0. Moreover in this case h is not an inner derivation of R; 4. there exist a′, c′ 2 U such that F(x) = a′x, G(x) = c′x, for all x ε R, with a′c′ = 0; 5. there exist b′, q′ 2 U such that F(x) = xb′, G(x) = xq′, for all x ε R, with q′b′ = 0; 6. there exist c′, q′ 2 U, η, γ ε C such that F(x) = η(xq′ -c′x)+γx, G(x) = c′x + xq′, for all x 2 R, with γc′ - ηc′2 = -γq′ - ηq′2.

### Composition of generalized derivations as a lie derivation

#### Abstract

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds: 1. there exist α 2 C and a 2 U such that F(x) = [a, x] and G(x) =αx, for all x εR; 2. G is an usual derivation of R and there exists α 2 C such that F(x) = αx, for all x ε R; 3. there exist α, β 2 C and a derivation h of R such that F(x) =αx+h(x), G(x) = βx, for all x ε R, and αβ+h(β) = 0. Moreover in this case h is not an inner derivation of R; 4. there exist a′, c′ 2 U such that F(x) = a′x, G(x) = c′x, for all x ε R, with a′c′ = 0; 5. there exist b′, q′ 2 U such that F(x) = xb′, G(x) = xq′, for all x ε R, with q′b′ = 0; 6. there exist c′, q′ 2 U, η, γ ε C such that F(x) = η(xq′ -c′x)+γx, G(x) = c′x + xq′, for all x 2 R, with γc′ - ηc′2 = -γq′ - ηq′2.
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2012
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11570/2436021`
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