Derivations with Central Values on Multilinear Polynomials in Prime Rings

2012 ◽  
Vol 4 (3) ◽  
pp. 85-93
Author(s):  
Dhara
Keyword(s):  
Prime Rings ◽  
Central Values ◽  
Multilinear Polynomials

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}}.

b-Generalized Derivations Acting on Multilinear Polynomials in Prime Rings

2018 ◽  
Vol 25 (04) ◽  
pp. 681-700
Author(s):  
Basudeb Dhara ◽  
Vincenzo De Filippis
Keyword(s):  
Prime Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Ring Of Quotients ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, Q be its maximal right ring of quotients, and C be its extended centroid. Suppose that [Formula: see text] is a non-central multilinear polynomial over C, [Formula: see text], and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case [Formula: see text] for all [Formula: see text] in Rn.

Actions of Generalized Derivations on Multilinear Polynomials in Prime Rings

2011 ◽  
Vol 18 (spec01) ◽  
pp. 955-964 ◽  
Author(s):  
Nurcan Argaç ◽  
Vincenzo De Filippis
Keyword(s):  
Commutative Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Standard Identity ◽  
Multilinear Polynomials

Let K be a commutative ring with unity, R a non-commutative prime K-algebra with center Z(R), U the Utumi quotient ring of R, C=Z(U) the extended centroid of R, I a non-zero two-sided ideal of R, H and G non-zero generalized derivations of R. Suppose that f(x1,…,xn) is a non-central multilinear polynomial over K such that H(f(X))f(X)-f(X)G(f(X))=0 for all X=(x1,…,xn)∈ In. Then one of the following holds: (1) There exists a ∈ U such that H(x)=xa and G(x)=ax for all x ∈ R. (2) f(x1,…,xn)2 is central valued on R and there exist a, b ∈ U such that H(x)=ax+xb and G(x)=bx+xa for all x ∈ R. (3) char (R)=2 and R satisfies s4, the standard identity of degree 4.

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