Generalized skew derivations with power central values on Lie ideals. (English) Zbl 1242.16038
Let \(R\) be a prime ring with center \(Z\) and extended centroid \(C\), and let \(\beta\) be an automorphism of \(R\). An additive map \(\delta\colon R\to R\) is called a \(\beta\)-derivation if \(\delta(xy)=\delta(x)y+\beta(x)\delta(y)\) for all \(x,y\in R\); an additive map \(f\colon R\to R\) is called a right generalized \(\beta\)-derivation if there exists a \(\beta\)-derivation \(\delta\) such that \(f(xy)=f(x)y+\beta(x)\delta(y)\) for all \(x,y\in R\).
It is proved that if \(L\) is a noncommutative Lie ideal of \(R\) and \(f\) is a right generalized \(\beta\)-derivation such that \(f(x)^n\in Z\) for all \(x\in L\) and some fixed positive integer \(n\), then \(f=0\) or \(\dim_CRC=4\). This result generalizes several known results for derivations, which are discussed in the introduction.
It is proved that if \(L\) is a noncommutative Lie ideal of \(R\) and \(f\) is a right generalized \(\beta\)-derivation such that \(f(x)^n\in Z\) for all \(x\in L\) and some fixed positive integer \(n\), then \(f=0\) or \(\dim_CRC=4\). This result generalizes several known results for derivations, which are discussed in the introduction.
Reviewer: Howard E. Bell (St. Catharines)
MSC:
| 16W25 | Derivations, actions of Lie algebras |
| 16N60 | Prime and semiprime associative rings |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 16R60 | Functional identities (associative rings and algebras) |
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