On Identities with Composition of Generalized Derivations

2017 ◽  
Vol 60 (4) ◽  
pp. 721-735 ◽  
Author(s):  
Münevver Pınar Eroglu ◽  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Central Closure

AbstractLet R be a prime ring with extended centroid C, Q maximal right ring of quotients of R, RC central closure of R such that dim C(RC) > , ƒ (X1, . . . , Xn) a multilinear polynomial over C that is not central-valued on R, and f (R) the set of all evaluations of the multilinear polynomial f (X1 , . . . , Xn) in R. Suppose that G is a nonzero generalized derivation of R such that G2(u)u ∈ C for all u ∈ ƒ(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.

scholarly journals Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
V. De Filippis ◽  
G. Scudo ◽  
L. Sorrenti
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Utumi Quotient Ring ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R, f(x1,…,xn) a noncentral multilinear polynomial over C in n noncommuting variables, and a,b∈R such that a[F(f(r1,…,rn)),f(r1,…,rn)]b=0 for any r1,…,rn∈R. Then one of the following holds: (1) a=0; (2) b=0; (3) there exists λ∈C such that F(x)=λx, for all x∈R; (4) there exist q∈U and λ∈C such that F(x)=(q+λ)x+xq, for all x∈R, and f(x1,…,xn)2 is central valued on R; (5) there exist q∈U and λ,μ∈C such that F(x)=(q+λ)x+xq, for all x∈R, and aq=μa, qb=μb.

scholarly journals Centralizing b-generalized derivations on multilinear polynomials

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6251-6266
Author(s):  
S.K. Tiwari ◽  
B. Prajapati
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Utumi Quotient Ring ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2 and F a b-generalized derivation on R. Let U be Utumi quotient ring of R with extended centroid C and f (x1,..., xn) be a multilinear polynomial over C which is not central valued on R. Suppose that d is a non zero derivation on R such that d([F(f(r)), f(r)]) ? C for all r = (r1,..., rn) ? Rn, then one of the following holds: (1) there exist a ? U, ? ? C such that F(x) = ax + ?x + xa for all x ? R and f (x1,..., xn)2 is central valued on R, (2) there exists ? ? C such that F(x) = ?x for all x ? R.

Generalized Derivations Having the Same Power Values with Left Multiplications

2013 ◽  
Vol 20 (03) ◽  
pp. 369-382 ◽  
Author(s):  
Xiaowei Xu ◽  
Jing Ma ◽  
Fengwen Niu
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Fixed Positive Integer

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal [Formula: see text] and a noncommutative Lie ideal L, respectively.

Annihilator Conditions with Generalized Derivations on Multilinear Polynomials

2013 ◽  
Vol 20 (04) ◽  
pp. 613-622
Author(s):  
Yiqiu Du ◽  
Yu Wang
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Idempotent Element ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Utumi Quotient Ring ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f(x1,…,xn) a multilinear polynomial over C, a ∈ R, and I a nonzero right ideal of R. Suppose that a[g(f(r1,…,rn)), f(r1,…,rn)]=0 for all ri∈ I and aI ≠ 0. Then either g(x)=a1x with (a1-γ)I=0 for some a1∈ U and γ ∈ C, or there exists an idempotent element e ∈ soc (RC) such that IC=eRC and one of the following holds: (i) f(x1,…,xn) is central-valued in eRe; (ii) g(x)=bx+xc, where b, c ∈ U with (c-b-α)e=0 for some α ∈ C and f(x1,…,xn) is central-valued in eRe.

Engel type identities with generalized derivations in prime rings

2018 ◽  
Vol 11 (04) ◽  
pp. 1850055
Author(s):  
Basudeb Dhara ◽  
Krishna Gopal Pradhan ◽  
Shailesh Kumar Tiwari
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Fixed Integer ◽  
Derivation Formula ◽  
Ring Of Quotients ◽  
Outer Derivation

Let [Formula: see text] be a noncommutative prime ring with its Utumi ring of quotients [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] a generalized derivation of [Formula: see text] and [Formula: see text] a nonzero ideal of [Formula: see text]. If [Formula: see text] satisfies any one of the following conditions: (i) [Formula: see text], [Formula: see text], [Formula: see text], (ii) [Formula: see text], where [Formula: see text] is a fixed integer, then one of the following holds: (1) there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (3) char [Formula: see text], [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and an outer derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text].

Identities with Product of Generalized Derivations of Prime Rings

2013 ◽  
Vol 20 (04) ◽  
pp. 711-720 ◽  
Author(s):  
Luisa Carini ◽  
Vincenzo De Filippis ◽  
Giovanni Scudo
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, f(x1,…,xn) a multilinear polynomial over C which is not an identity for R, F and G two non-zero generalized derivations of R. If F(u)G(u)=0 for all u ∈ f(R)= {f(r1,…,rn): ri∈ R}, then one of the following holds: (i) There exist a, c ∈ U such that ac=0 and F(x)=xa, G(x)=cx for all x ∈ R; (ii) f(x1,…,xn)2is central valued on R and there exist a, c ∈ U such that ac=0 and F(x)=ax, G(x)=xc for all x ∈ R; (iii) f(x1,…,xn) is central valued on R and there exist a,b,c,q ∈ U such that (a+b)(c+q)=0 and F(x)=ax+xb, G(x)=cx+xq for all x ∈ R.

scholarly journals Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Basudeb Dhara ◽  
Shervin Sahebi ◽  
Venus Rahmani
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Jordan Homomorphisms

AbstractLet R be a prime ring with center Z(R) and extended centroid C, H a non-zero generalized derivation of R and n ≥ 1 a fixed integer. In this paper we study the situations:(1) H(u(2) H(u

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.

scholarly journals Extended centroids of power series rings

1990 ◽  
Vol 32 (3) ◽  
pp. 371-375 ◽  
Author(s):  
W. S. Martindale ◽  
M. P. Rosen ◽  
J. D. Rosen
Keyword(s):  
Prime Ring ◽  
Polynomial Identity ◽  
Extended Centroid ◽  
Prime Rings ◽  
Generalized Polynomial ◽  
Ring Of Quotients ◽  
Power Series Rings ◽  
Central Simple ◽  
Generalized Polynomial Identity ◽  
Central Closure

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.

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