Let R be a ring, Qr the right Martindale quotient ring of R, C the extended centroid of R, L a noncentral Lie ideal of R, F a nonzero generalized skew derivation of R, and m, n, k≥ 1 be fixed integers. We prove the following results:(a)Assume R is a prime. If either char(R) = 0 or char(R) > m+ 1 and [F(um),un]k=0 for all u∈ L, then there exists λ∈ C such that F(x) = λx, for all x∈ R, unless when R satisfies s4, the standard identity of degree 4.(b)Assume R is semiprime. If char(R) ≠ 2 and [F(x),xn]k=0 for all x∈ R, then either there exists λ∈ C such that F(x) = λx, for all x∈ R, or R contains a non-zero central ideal. © 2018, Iranian Mathematical Society.
Prime and Semiprime Rings with n-Commuting Generalized Skew Derivations
De Filippis, V.
2018-01-01
Abstract
Let R be a ring, Qr the right Martindale quotient ring of R, C the extended centroid of R, L a noncentral Lie ideal of R, F a nonzero generalized skew derivation of R, and m, n, k≥ 1 be fixed integers. We prove the following results:(a)Assume R is a prime. If either char(R) = 0 or char(R) > m+ 1 and [F(um),un]k=0 for all u∈ L, then there exists λ∈ C such that F(x) = λx, for all x∈ R, unless when R satisfies s4, the standard identity of degree 4.(b)Assume R is semiprime. If char(R) ≠ 2 and [F(x),xn]k=0 for all x∈ R, then either there exists λ∈ C such that F(x) = λx, for all x∈ R, or R contains a non-zero central ideal. © 2018, Iranian Mathematical Society.Pubblicazioni consigliate
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