scholarly journals ON mTH-COMMUTATORS AND ANTI-COMMUTATORS INVOLVING GENERALIZED DERIVATIONS IN PRIME RINGS

2019 ◽  
pp. 391
Author(s):  
Mohd Arif Raza
Keyword(s):  
Quotient Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Specific Element ◽  
Utumi Quotient Ring

In this manuscript, we study the $m$-th commutator and anti-commutator involving generalized derivations on some suitable subsets of rings. We attain the information about the structure of rings and the behaviour of generalized derivation in form of multiplication by some specific element of Utumi quotient ring which satisfies certain differential identities.

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.

Prime Rings with Generalized Derivations on Right Ideals

2011 ◽  
Vol 18 (spec01) ◽  
pp. 987-998 ◽  
Author(s):  
Ç. Demir ◽  
N. Argaç
Keyword(s):  
Commutative Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Standard Identity

Let K be a commutative ring with unit, R be a prime K-algebra with center Z(R), right Utumi quotient ring U and extended centroid C, and I a nonzero right ideal of R. Let g be a nonzero generalized derivation of R and f(X1,…,Xn) a multilinear polynomial over K. If g(f(x1,…,xn)) f(x1,…,xn) ∈ C for all x1,…,xn ∈ I, then either f(x1,…,xn)xn+1 is an identity for I, or char (R)=2 and R satisfies the standard identity s4(x1,…,x4), unless when g(x)=ax+[x,b] for suitable a, b ∈ U and one of the following holds: (i) a, b ∈ C and f(x1,…,xn)2 is central valued on R; (ii) a ∈ C and f(x1,…,xn) is central valued on R; (iii) aI=0 and [f(x1,…,xn), xn+1]xn+2 is an identity for I; (iv) aI=0 and (b-β)I=0 for some β ∈ C.

Some results on ∗-differential identities in prime rings

2021 ◽  
Author(s):  
Deepak Kumar ◽  
Bharat Bhushan ◽  
Gurninder S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Derivation Formula

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

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 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.

Actions of Generalized Derivations on Multilinear Polynomials in Prime Rings

2011 ◽  
Vol 18 (spec01) ◽  
pp. 955-964 ◽  
Author(s):  
Nurcan Argaç ◽  
Vincenzo De Filippis
Keyword(s):  
Commutative Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Standard Identity ◽  
Multilinear Polynomials

Let K be a commutative ring with unity, R a non-commutative prime K-algebra with center Z(R), U the Utumi quotient ring of R, C=Z(U) the extended centroid of R, I a non-zero two-sided ideal of R, H and G non-zero generalized derivations of R. Suppose that f(x1,…,xn) is a non-central multilinear polynomial over K such that H(f(X))f(X)-f(X)G(f(X))=0 for all X=(x1,…,xn)∈ In. Then one of the following holds: (1) There exists a ∈ U such that H(x)=xa and G(x)=ax for all x ∈ R. (2) f(x1,…,xn)2 is central valued on R and there exist a, b ∈ U such that H(x)=ax+xb and G(x)=bx+xa for all x ∈ R. (3) char (R)=2 and R satisfies s4, the standard identity of degree 4.

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 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.

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.

Higher-order commutators with power central values on rings and algebras involving generalized derivations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shakir Ali ◽  
Husain Alhazmi ◽  
Abdul Nadim Khan ◽  
Mohd Arif Raza
Keyword(s):  
Quotient Ring ◽  
Generalized Derivation ◽  
Higher Order ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Left Multiplication ◽  
Central Values ◽  
Power Central ◽  
Rings And Algebras

AbstractLet {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

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