Power-Central Values and Engel Conditions in Prime Rings with Generalized Skew Derivations

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
N. Argaç ◽  
V. De Filippis
Keyword(s):  
Prime Rings ◽  
Central Values ◽  
Power Central

scholarly journals Derivations with power central values on Lie ideals in prime rings

2008 ◽  
Vol 58 (1) ◽  
pp. 147-153 ◽  
Author(s):  
Basudeb Dhara ◽  
R. K. Sharma
Keyword(s):  
Prime Rings ◽  
Central Values ◽  
Power Central

Higher-order commutators with power central values on rings and algebras involving generalized derivations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shakir Ali ◽  
Husain Alhazmi ◽  
Abdul Nadim Khan ◽  
Mohd Arif Raza
Keyword(s):  
Quotient Ring ◽  
Generalized Derivation ◽  
Higher Order ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Left Multiplication ◽  
Central Values ◽  
Power Central ◽  
Rings And Algebras

AbstractLet {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

scholarly journals Commutators with power central values on a Lie ideal

2000 ◽  
Vol 193 (2) ◽  
pp. 269-278 ◽  
Author(s):  
Luisa Carini ◽  
Vincenzo De Filippis
Keyword(s):  
Lie Ideal ◽  
Central Values ◽  
Power Central

Generalized Derivations with Power-Central Values on Multilinear Polynomials

2006 ◽  
Vol 13 (03) ◽  
pp. 405-410 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Commutative Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Algebra ◽  
Standard Identity ◽  
Central Values ◽  
Multilinear Polynomials ◽  
Power Central

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X1, …,Xt) a multilinear polynomial over K. If g(f (X1, …,Xt))n ∈ Z for all x1, …, xt ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x1, …,xt) is central-valued on R except when R satisfies s4, the standard identity in four variables.

Skew Derivations with Power Central Values on Commutators

2015 ◽  
Vol 22 (03) ◽  
pp. 479-494 ◽  
Author(s):  
Hung-Yuan Chen
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Prime Ring ◽  
Skew Derivation ◽  
Central Values ◽  
Fixed Positive Integer ◽  
Power Central

Let R be a prime ring with center [Formula: see text], δ: R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if (i) [δ([x,y]),[x,y]]n = 0 for all x, y ∈ R or (ii) [Formula: see text] for all x ∈ R, except some specific cases.

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