Search

In this paper, we investigate the commutativity of a prime Banach algebra 𝔄 which admits a nonzero continuous linear generalized derivation 𝔉 : 𝔄 → 𝔄 associated with continuous linear derivation δ such that either 𝔉 ([ 𝔵 m , 𝔶 n ]) − [ 𝔉 (𝔵 m) , 𝔶 n ] ∈ 𝒵 (𝔄) or 𝔉 ([ 𝔵 m , 𝔶 n ]) + [ 𝔉 (𝔵 m) , 𝔶 n ] ∈ 𝒵 (𝔄) for intergers m = m (𝔵 , 𝔶) > 1 and n = n (𝔵 , 𝔶) > 1 and sufficiently many 𝔵 , 𝔶 ∈ 𝔄. Further, similar results are also obtained for unital prime Banach algebra 𝔄 which admits a nonzero continuous linear generalized derivations 𝔉 satisfying either 𝔉 (𝔵 𝔶) n − 𝔉 (𝔵 n) 𝔶 n ∈ 𝒵 (𝔄) or 𝔉 (𝔵 𝔶) n − 𝔶 n 𝔉 (𝔵 n) ∈ 𝒵 (𝔄) for an integer n = n (𝔵 , 𝔶) > 1 and sufficiently many 𝔵 , 𝔶. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f (x1, . . . , xn) a multilinear polynomial over C, ρ a nonzero right ideal of R andm > 1 a fixed integer such that ([d(f(r1, . . . , rn)), f(r1, . . . , rn)])m = [d(f(r1, . . . , rn)), f(r1, . . . , rn)] for all r1, . . . , rn ∈ ρ. Then either [f(x1,..., xn), xn+1]xn+2 is an identity for ρ or d(ρ)ρ = 0. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x), d(y)]m = [x, y] for all x, y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer--Werner theorem. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F;G;H be the generalized derivations with associated derivations d,δh of R respectively. In the present paper, we study the situations if one the follwoing holds (1) F(u)G(v)±H(uv)ϵ Z(R), (2) F(u)F (v)±H(vu) ϵ Z(R), for all u,v ϵ U, then U ⊆Z(R). [ABSTRACT FROM AUTHOR]

In this paper we extend to the higher homomorphism a well known result proved by Herstein concerning homomorphism in prime rings. We prove results which imply that every Jordan triple higher homomorphism (resp. generalized Jordan triple higher homomorphism) of a ring R onto a 2-torsion-free prime rings R' is of the form Φi = ±Ψi for all i ∈ ..., where Ψi ∈ Ψ = (Ψi)i∈N which Ψ is either higher homomorphism or higher anti-homomorphism (resp. Ψ is either generalized higher homomorphism or generalized higher anti-homomorphism) of R onto R'. [ABSTRACT FROM AUTHOR]

Let R be a *-prime ring with characteristic different from two and U ≠ 0 be a square closed *-Lie ideal of R. An additive mapping F : R → R is called an generalized derivation if there exits a derivation d : R → R such that F(xy) = F(x)y+xd(y). In the present paper, it is shown that U ⊆ Z if R is a *-prime ring which admits a generalized derivation satisfying several conditions that are associated with a derivation commuting with *. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 with extended centroid C. Let F be a generalized derivation of R, L a non-central Lie ideal of R, f(x1, ..., xn) a polynomial over C and f(R)={f(r1, ..., rn): ri ∈ R}. We study the following cases: (1) [F(u), F(v)]k=0 for all u, v ∈ L, where k ≥ 1 is a fixed integer; (2) [F(u), F(v)] = 0 for all u, v ∈ f(R); (3) F(u) ◦ F(v)=0 for all u, v ∈ f(R); (4) F(u) ◦ F(v)=u ◦ v for all u, v ∈ f(R). We obtain a description of the structure of R and information on the form of F. [ABSTRACT FROM AUTHOR]

Let R be a ring and α,β be endomorphisms of R. An additive mapping F: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d: R → R such that F(xy)=F(x) α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]α,β, (v) F(x ◦ y)=(x ◦ y)α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R. [ABSTRACT FROM AUTHOR]