Product of Generalized Derivations with Commuting Values on a Lie Ideal

2020 ◽  
pp. 123-135
Author(s):  
Luisa Carini ◽  
Vincenzo De Filippis ◽  
Giovanni Scudo
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations

scholarly journals Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem UR Rehman ◽  
Abu Zaid Ansari
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers ◽  
Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

scholarly journals Posner’s second theorem and some related annihilating conditions on lie ideals

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1285-1301 ◽  
Author(s):  
Emine Albaş ◽  
Nurcan Argaç ◽  
Filippis de
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
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, L a non-central Lie ideal of R, F and G two non-zero generalized derivations of R. If [F(u),u]G(u) = 0 for all u ? L, then one of the following holds: (a) there exists ? ? C such that F(x) = ?x, for all x ? R; (b) R ? M2(F), the ring of 2 x 2 matrices over a field F, and there exist a ? U and ? ? C such that F(x) = ax + xa + ?x, for all x ? R.

Generalized derivations preserving Engel condition in prime and semiprime rings

2020 ◽  
pp. 2150041
Author(s):  
Rita Prestigiacomo
Keyword(s):  
Prime Ring ◽  
Algebraic Closure ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Martindale Quotient Ring ◽  
Engel Condition

Let [Formula: see text] be a prime ring with [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] its Martindale quotient ring and [Formula: see text] its extended centroid. Let [Formula: see text] and [Formula: see text] be nonzero generalized derivations on [Formula: see text] such that [Formula: see text] Then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text], for any [Formula: see text], unless [Formula: see text], where [Formula: see text] is the algebraic closure of [Formula: see text].

Identities with Generalized Derivations on Prime Rings and Banach Algebras

2012 ◽  
Vol 19 (spec01) ◽  
pp. 971-986 ◽  
Author(s):  
Luisa Carini ◽  
Vincenzo De Filippis
Keyword(s):  
Banach Algebras ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Algebraic Result ◽  
Spectrally Bounded ◽  
Inclusion Results

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, and H, G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(un)un + unG(un) ∈ C for all u ∈ L, then either there exists a ∈ U such that H(x) = xa and G(x) = -ax, or R satisfies the standard identity s4 and one of the following holds: (i) char (R) = 2; (ii) n is even and there exist a′ ∈ U, α ∈ C and derivations d, δ of R such that H(x) = a′ x + d(x) and G(x) = (α-a′)x + δ(x); (iii) n is even and there exist a′ ∈ U and a derivation δ of R such that H(x)=xa′ and G(x) = -a′ x + δ(x); (iv) n is odd and there exist a′, b′ ∈ U and α, β ∈ C such that H(x) = a′ x + x(β-b′) and G(x) = b′ x+x(α-a′); (v) n is odd and there exist α, β ∈ C and a derivation d of R such that H(x) = α x+d(x) and G(x) = β x + d(x); (vi) n is odd and there exist a′ ∈ U and α ∈ C such that H(x) = xa′ and G(x) = (α - a′)x. As an application of this purely algebraic result, we obtain some range inclusion results of continuous or spectrally bounded generalized derivations H and G on Banach algebras R satisfying the condition H(xn)xn + xnG(xn) ∈ rad (R) for all x ∈ R, where rad (R) is the Jacobson radical of R.

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 Some results on generalized derivations and (σ,τ)-Lie ideals

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5527-5535
Author(s):  
Evrim Güven
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Generalized Derivations

Let R be a prime ring with characteristic not 2 and ?,?,?,?,?,?,? automorphisms of R. Let h : R ? R be a nonzero left(resp.right)-generalized (?,?)-derivation, b ? R and V ? 0 a left (?,?)-Lie ideal of R. The main object in this article is to study the situations. (1) h(I) ? C?,?(V),(2) bh(I) ? C?,?(V) or h(I)b ? C?,?0V), (3) h?(V)=0,(4) h?(V)b=0 or bh?(V)=0.

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]

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 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.

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

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