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

Commutativity and Skew-commutativity Conditions with Generalized Derivations

2010 ◽  
Vol 17 (spec01) ◽  
pp. 841-850 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem ur Rehman
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer

Let R be a prime ring of characteristic different from 2 with extended centroid C. Let F be a generalized derivation of R, L a non-central Lie ideal of R, f(x1, …, xn) a polynomial over C and f(R)={f(r1, …, rn): ri ∈ R}. We study the following cases: (1) [F(u), F(v)]k=0 for all u, v ∈ L, where k ≥ 1 is a fixed integer; (2) [F(u), F(v)] = 0 for all u, v ∈ f(R); (3) F(u) ◦ F(v)=0 for all u, v ∈ f(R); (4) F(u) ◦ F(v)=u ◦ v for all u, v ∈ f(R). We obtain a description of the structure of R and information on the form of F.

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].

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.

scholarly journals A pair of generalized derivations in prime, semiprime rings and in Banach algebras

2021 ◽  
Vol 39 (4) ◽  
pp. 131-141
Author(s):  
Basudeb Dhara ◽  
Venus Rahmani ◽  
Shervin Sahebi
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Banach Algebras ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Semiprime Rings ◽  
Commutative Banach Algebras

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy)+G(yx))n= (xy ±yx) for all x,y ∈ I, then one ofthe following holds:(1) R is commutative;(2) n = 1 and H(x) = x and G(x) = ±x for all x ∈ R.Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

scholarly journals ANNIHILATORS OF POWER VALUES OF GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS

2009 ◽  
Vol 80 (2) ◽  
pp. 217-232 ◽  
Author(s):  
VINCENZO DE FILIPPIS
Keyword(s):  
Quotient Ring ◽  
Generalized Derivation ◽  
List Type ◽  
Type Number ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Utumi Quotient Ring ◽  
Fixed Element ◽  
Multilinear Polynomials

AbstractLet R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1,…,xn) be a noncentral multilinear polynomial over C, m≥1 a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag(f(r1,…,rn))m=0 for all r1,…,rn∈I, then one of the following holds: (1)aI=ag(I)=(0);(2)g(x)=qx, for some q∈U and aqI=0;(3)[f(x1,…,xn),xn+1]xn+2 is an identity for I;(4)g(x)=cx+[q,x] for all x∈R, where c,q∈U such that cI=0 and [q,I]I=0.

scholarly journals Generalized Derivations of Prime Gamma Rings

2014 ◽  
Vol 33 ◽  
pp. 33-39
Author(s):  
Kalyan Kumar Dey ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Gamma Rings

Let M be a prime ?-ring satisfying a certain assumption a?b?c = a?b?c for all a, b, c?M and ?, ???, and let I be an ideal of M. Assume that (D, d) is a generalized derivation of M and a?M. If D([x, a]?) = 0 or [D(x), a]? = 0 for all x?I, ? ? ?, then we prove that d(x) = p?[x, a]? for all x?I, ?, ? ? ? or a?Z(M) (the centre of M), where p belongs C(M) (the extended centroid of M). GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 33-39 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17654

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).

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.

Generalized Derivations on Lie Ideals and Power Values on Prime Rings

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Giovanni Scudo ◽  
Abu Zaid Ansari
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Standard Identity

AbstractLet R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.If [G(u), u](1) R satisfies the standard identity s(2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.

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