Power Values of Generalized Skew Derivations with Annihilator Conditions on Lie Ideals
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Power Values of Generalized Skew Derivations with Annihilator Conditions on Lie Ideals Münevver Pınar Eroglu ˘ 1
· Nurcan Argaç2
Received: 6 August 2018 / Revised: 8 November 2019 / Accepted: 20 December 2019 © Iranian Mathematical Society 2020
Abstract Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations δ and of R in case of p(δ(u)l1 (u)l2 δ(u)l3 (u)l4 · · · (u)lk )n = 0, for all u ∈ L, where 0 = p ∈ R, l1 , l2 , . . . , lk are fixed nonnegative integers with l1 = 0 and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if pδ(u)n = 0 for all u ∈ L, then there exists a ∈ Q such that δ(x) = ax for all x ∈ R and pa = 0 unless R satisfies s4 . Keywords Prime ring · Maximal right rings of quotients · Extended Jacobson density theorem · Generalized skew derivation · Automorphism Mathematics Subject Classification 16N60 · 16R50 · 16W20 · 16W25 · 16W55
1 Introduction and Results Posner [23] proved that if ad(x) = 0 for all x ∈ R, then either a = 0 or d = 0 where R is a prime ring with a derivation d and a ∈ R. The results have led to many works on
Communicated by Mohammad Saeid Azam.
B
Münevver Pınar Ero˘glu [email protected] Nurcan Argaç [email protected]
1
Department of Mathematics, Science Faculty, Dokuz Eylül University, Izmir, Turkey
2
Department of Mathematics, Science Faculty, Ege University, Izmir, Turkey
123
Bulletin of the Iranian Mathematical Society
power values of derivations and generalized derivations with annihilator condition in prime rings [3,12,18,24]. Breˇsar [3] extended the Posner’s result to power value case and proved that if R is a (n − 1)!-torsion free semiprime ring with a derivation d, n is a positive fixed integer and a ∈ R such that ad(x)n = 0 for all x ∈ R, then ad(R) = 0. Moreover, if R is a prime ring, then either a = 0 or d = 0. Lee and Lin [18] extended this result to Lie ideal case without any restriction on torsion free of R and proved that if ad(u)n = 0 for all u ∈ L, then ad(L) = 0 unless char (R) = 2 and dimC RC = 4, where R is a prime ring with a derivation d and a Lie ideal L, n is a positive fixed integer and a ∈ R. Also it is proved that if L is noncommutative, then ad(R) = 0, i.e., a = 0 or d = 0. Chang [7] generalized the result of Breˇsar [3] to generalized skew derivations and proved that if R is a prime ring with a generalized σ -derivation δ, n is a positive fixed integer and a ∈ R such that aδ(x)n = 0 for all x ∈ R; then aδ(R) = 0. In [25], the second author and Yarbil extended this result to the Lie ideal case under the same conditions and proved that if aδ(u)n = 0 for all u ∈ L where L is a noncommutative Lie ideal of R, then aδ(R) = 0 unless dim C (RC) = 4. Recently in [24], Shutaj and Khan proved that if R is a prime ring with a generalize
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