Derivations with annihilator conditions on Lie ideals in prime rings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050025 ◽  
Author(s):  
Shuliang Huang
Keyword(s):  
Prime Ring ◽  
Polynomial Identity ◽  
Lie Ideal ◽  
Prime Rings ◽  
Positive Integers

Let [Formula: see text] be a prime ring with characteristic different from two, [Formula: see text] a derivation of [Formula: see text], [Formula: see text] a noncentral Lie ideal of [Formula: see text], and [Formula: see text]. In the present paper, it is shown that if one of the following conditions holds: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text] and (iv) [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then [Formula: see text] unless [Formula: see text] satisfies [Formula: see text], the standard polynomial identity in four variables.

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 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.

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.

scholarly journals Remarks on *−(σ,Τ)− Lie Ideals of *−Prime Rings with Derivation

2018 ◽  
Vol 60 (1) ◽  
pp. 161-171
Author(s):  
Emine K. Sögütcü ◽  
Neşet Aydin ◽  
Öznur Gölbaşi
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings

Abstract Let R be a ∗−prime ring with characteristic not 2, U a nonzero ∗− (σ,τ)−Lie ideal of R, d a nonzero derivation of R. Suppose σ, τ be two automorphisms of R such that σd = dσ, τd = dτ and ∗ commutes with σ, τ, d. In the present paper it is shown that if d(U) ⊆ Z or d2(U) ⊆ Z, then U ⊆ Z.

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]

Centralizing Automorphisms of Lie Ideals in Prime Rings

1992 ◽  
Vol 35 (4) ◽  
pp. 510-514 ◽  
Author(s):  
Joseph H. Mayne
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings

AbstractLet R be a prime ring of characteristic not equal to two and let T be an automorphism of R. If U is a Lie ideal of R such that T is nontrivial on U and xxT — xTx is in the center of R for every x in U, then U is contained in the center of R.

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.

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

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.

A note on automorphisms of lie ideals in prime rings

2018 ◽  
Vol 68 (5) ◽  
pp. 1223-1229 ◽  
Author(s):  
Bijan Davvaz ◽  
Mohd Arif Raza
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings ◽  
Standard Identity ◽  
Identity Automorphism ◽  
Fixed Positive Integer

Abstract In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard identity in four variables if R admits a non-identity automorphism σ such that (uσ,u]vσ+vσ[uσ,u])n∈Z for all u,v in some non-central Lie ideal L of R whenever either char(R)>n or char(R)=0, where n is a fixed positive integer.

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