Generalized derivations vanishing on co-commutator identities in prime rings

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]).

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

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.

Centralizing b-generalized derivations on 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.

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

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 on Lie Ideals and Power Values on Prime Rings

Let 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.

Two-Sided Annihilator Condition with Generalized Derivations on 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.

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

Let 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

Identities with Product of Generalized Derivations of Prime Rings

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.

Annihilator Conditions with Generalized Derivations on 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.

Identities with Generalized Derivations on Prime Rings and Banach Algebras

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.