Derivations with power central values on multilinear polynomials in prime rings

2008 ◽  
Vol 57 (3) ◽  
pp. 401-410
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Rings ◽  
Central Values ◽  
Multilinear Polynomials ◽  
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

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.

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.

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

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