Generalized derivations on Lie ideals in prime rings

2016 ◽  
Vol 10 (02) ◽  
pp. 1750032 ◽  
Author(s):  
V. K. Yadav ◽  
S. K. Tiwari ◽  
R. K. Sharma
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Square Closed Lie Ideal ◽  
Torsion Free

Let [Formula: see text] be a [Formula: see text]-torsion free prime ring, and [Formula: see text] a square closed Lie ideal of [Formula: see text] Further let [Formula: see text] and [Formula: see text] be generalized derivations associated with derivations [Formula: see text] and [Formula: see text], respectively on [Formula: see text] If one of the following conditions holds: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] (v) [Formula: see text] for all [Formula: see text] then it is proved that either [Formula: see text] or [Formula: see text]

scholarly journals Generalized Derivations Acting as Homomorphisms and Anti-Homomorphisms on Lie Ideals of Prime Rings

2016 ◽  
Vol 35 ◽  
pp. 73-77
Author(s):  
Akhil Chandra Paul ◽  
Sujoy Chakraborty
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Square Closed Lie Ideal ◽  
Torsion Free

Let U be a non-zero square closed Lie ideal of a 2-torsion free prime ring R and f a generalized derivation of R with the associated derivation d of R. If f acts as a homomorphism and as an anti-homomorphism on U, then we prove that d = 0 or U € Z(R), the centre of R.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 73-77

scholarly journals Jordan Derivations on Lie Ideals of ?-Prime Rings

2017 ◽  
Vol 36 ◽  
pp. 1-5
Author(s):  
Akhil Chandra Paul ◽  
Md Mizanor Rahman
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Lie Ideal ◽  
Prime Rings ◽  
R And D ◽  
Square Closed Lie Ideal ◽  
Torsion Free

In this paper we prove that, if U is a s-square closed Lie ideal of a 2-torsion free s-prime ring R and  d: R(R is an additive mapping satisfying d(u2)=d(u)u+ud(u) for all u?U then d(uv)=d(u)v+ud(v) holds for all  u,v?UGANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 1-5

scholarly journals Derivations on Lie Ideals of σ-Prime Γ-Rings

2015 ◽  
Vol 39 (2) ◽  
pp. 249-255
Author(s):  
Md Mizanor Rahman ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Lie Ideal ◽  
Square Closed Lie Ideal ◽  
Academy Of Sciences ◽  
Gamma Rings ◽  
Torsion Free

The authors extend and generalize some results of previous workers to ?-prime ?-ring. For a ?-square closed Lie ideal U of a 2-torsion free ?-prime ?-ring M, let d: M ?M be an additive mapping satisfying d(u?u)=d(u)? u + u?d(u) for all u ? U and ? ? ?. The present authors proved that d(u?v) = d(u)?v + u?d(v) for all u, v ? U and ?? ?, and consequently, every Jordan derivation of a 2-torsion free ?-prime ?-ring M is a derivation of M.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 2, 249-255, 2015

scholarly journals Generalized Derivations of Prime Rings

2007 ◽  
Vol 2007 ◽  
pp. 1-6 ◽  
Author(s):  
Huang Shuliang
Keyword(s):  
Prime Ring ◽  
Additive Function ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings

LetRbe an associative prime ring,Ua Lie ideal such thatu2∈Ufor allu∈U. An additive functionF:R→Ris called a generalized derivation if there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)holds for allx,y∈R. In this paper, we prove thatd=0orU⊆Z(R)if any one of the following conditions holds: (1)d(x)∘F(y)=0, (2)[d(x),F(y)=0], (3) eitherd(x)∘F(y)=x∘yord(x)∘F(y)+x∘y=0, (4) eitherd(x)∘F(y)=[x,y]ord(x)∘F(y)+[x,y]=0, (5) eitherd(x)∘F(y)−xy∈Z(R)ord(x)∘F(y)+xy∈Z(R), (6) either[d(x),F(y)]=[x,y]or[d(x),F(y)]+[x,y]=0, (7) either[d(x),F(y)]=x∘yor[d(x),F(y)]+x∘y=0for allx,y∈U.

Co-commutators with Generalized Derivations on Lie Ideals in Prime Rings

2013 ◽  
Vol 20 (04) ◽  
pp. 593-600 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Standard Identity

Let R be a prime ring of characteristic different from 2, L a noncentral Lie ideal of R, H and G two nonzero generalized derivations of R. Suppose us(H(u)u-uG(u)) ut=0 for all u ∈ L, where s, t ≥ 0 are fixed integers. Then either (i) there exists p ∈ U such that H(x)=xp for all x ∈ R and G(x)=px for all x ∈ R unless R satisfies S4, the standard identity in four variables; or (ii) R satisfies S4 and there exist p, q ∈ U such that H(x)=px+xq for all x ∈ R and G(x)=qx+xp for all x ∈ R.

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.

scholarly journals (U, M) -derivations in prime ?-rings

2016 ◽  
Vol 27 (2) ◽  
pp. 143-153
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings ◽  
Torsion Free

In this article, we define (U,M)-derivation d of a ? -ring M . For a Lie ideal U of a 2 - torsion free prime ? -ring M satisfying the condition a?b?c = a?b?c for all a,b, c?M and ? ,? ?? , we prove the following results:(i) ifU is an admissible Lie ideal of M, then d(u?v) = d(u)?v + u?d(v) for all u, v?U ,? ??(ii) if u?u?U for all u?U,? ?? , then d(u?m) = d(u)?m + u?d(m) for all m ? M Bangladesh J. Sci. Res. 27(2): 143-153, December-2014

Permuting tri-derivations in prime and semi-prime rings

2016 ◽  
Vol 5 (1) ◽  
pp. 52
Author(s):  
H. Durna ◽  
S. OĞUZ
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings ◽  
Square Closed Lie Ideal

Let \(R\) be a ring and \(U\neq0\) be a square closed Lie ideal of \(R\). A tri-additive permuting map \(D:R\times R\times R\rightarrow R\) is called permuting tri-derivation if, for any \(y,z\in R\), the map \(x\mapsto D(x,y,z)\) is a derivation. A mapping \(d:R\rightarrow R\) defined by \(d(x)=D(x,x,x)\) is called the trace of \(D\). In the present paper, we show that \(U\subseteq Z\) such that \(R\) is a prime and semi-prime ring admitting the trace $d$ satisfying the several conditions of permuting tri-derivation.

scholarly journals σ- ideals and generalized derivations in σ-prime rings

2013 ◽  
Vol 31 (2) ◽  
pp. 113
Author(s):  
M. Rais Khan ◽  
Deepa Arora ◽  
M. Ali Khan
Keyword(s):  
Prime Ring ◽  
Generalized Derivations ◽  
Prime Rings

Let R be a prime ring and F and G be generalized derivations of R with associated derivations d and g respectively. In the present paper, we shall investigate the commutativity of R admitting generalized derivations F and G satisfying any one of the properties: (i) F(x)x = x G(x), (ii) F(x2) = x2 , (iii) [F(x), y] = [x, G(y)], (iv) d(x)F(y) = xy, (v) F([x, y]) = [F(x), y] + [d(y), x] and (vi) F(x ◦ y) = F(x) ◦ y − d(y) ◦ x for all x, y in some appropriate subset of R.

scholarly journals Multiplicative (generalized)-derivations of prime rings that act as $n$-(anti)homomorphisms

2020 ◽  
Vol 53 (2) ◽  
pp. 125-133
Author(s):  
G.S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivations ◽  
Prime Rings ◽  
The Given

Let R be a prime ring. In this note, we describe the possible forms of multiplicative (generalized)-derivations of R that act as n-homomorphism or n-antihomomorphism on nonzero ideals of R. Consequently, from the given results one can easily deduce the results of Gusić ([7]).

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