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.

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.

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.

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.

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.

Generalized Derivations with Power-Central Values on Multilinear Polynomials

2006 ◽  
Vol 13 (03) ◽  
pp. 405-410 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Commutative Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Algebra ◽  
Standard Identity ◽  
Central Values ◽  
Multilinear Polynomials ◽  
Power Central

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X1, …,Xt) a multilinear polynomial over K. If g(f (X1, …,Xt))n ∈ Z for all x1, …, xt ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x1, …,xt) is central-valued on R except when R satisfies s4, the standard identity in four variables.

scholarly journals Cocentralizing Generalized Derivations On Multilinear Polynomial On Right Ideals Of Prime Rings

2014 ◽  
Vol 47 (1) ◽  
Author(s):  
Vincenzo De Filippis ◽  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring

AbstractLet R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R ƒ (x1… xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(ƒ (r)) ƒ (r)- ƒ (r)H(ƒ (r)) ∈ C, for all r =(r1. there exist a; b; p ∈ U and α C such that G(x)= ax + [p, x] and H(x) = bx, for all x ∈ R, and (a-b)I=(0)=(a + p- α)I;2. R satisfies s3. R satisfies s4. R satisfies s5. there exists e(a) [ƒ (x(b) char (R) = 2 and s(c) [ƒ (x

scholarly journals Generalized derivations vanishing on co-commutator identities in prime rings

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1785-1801
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring

Let R be a noncommutative prime ring of char (R)? 2 with Utumi quotient ring U and extended centroid C and I a nonzero two sided ideal of R. Suppose that F(? 0), G and H are three generalized derivations of R and f (x1,...,xn) is a multilinear polynomial over C, which is not central valued on R. If F(G(f(r))f(r)- f(r)H(f(r))) = 0 for all r = (r1,..., rn) ? In, then we obtain information about the structure of R and describe the all possible forms of the maps F, G and H. This result generalizes many known results recently proved by several authors ([1], [4], [5], [8], [9], [13], [15]).

Annihilator Conditions with Generalized Derivations on Multilinear Polynomials

2013 ◽  
Vol 20 (04) ◽  
pp. 613-622
Author(s):  
Yiqiu Du ◽  
Yu Wang
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Idempotent Element ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Utumi Quotient Ring ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f(x1,…,xn) a multilinear polynomial over C, a ∈ R, and I a nonzero right ideal of R. Suppose that a[g(f(r1,…,rn)), f(r1,…,rn)]=0 for all ri∈ I and aI ≠ 0. Then either g(x)=a1x with (a1-γ)I=0 for some a1∈ U and γ ∈ C, or there exists an idempotent element e ∈ soc (RC) such that IC=eRC and one of the following holds: (i) f(x1,…,xn) is central-valued in eRe; (ii) g(x)=bx+xc, where b, c ∈ U with (c-b-α)e=0 for some α ∈ C and f(x1,…,xn) is central-valued in eRe.

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.

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