On Lie ideals with generalized derivations

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∈ R and [a, f(U)] = 0 then a ∈ Z or d(a) = 0 or U ? Z; (ii) If f ²(U) = 0 then U ? Z; (iii) If u ² ∈ U for all u ∈ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ? Z.     

Derivations as Homomorphisms or Anti-homomorphisms on Lie Ideals

Let R be a 2-torsion free prime ring, Z the center of R, and U a nonzero Lie ideal of R. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U, then either d = 0 or U lohtein in Z. This result improves a theorem of Asma, Rehman, and Shakir.     

Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

A Note on Commutators with Power Central Values on Lie Ideals

This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.     

Derivations with algebraic values of bounded degree

Let Rbe a prime algebra over a field .F, d a nonzero derivation of Rand ρ a nonzero right ideal of R. Suppose that for every x∈ ρ,d(x) is algebraic over Fof bounded degree. Then Ris a primitive ring with a minimal right ideal eR, where e=e² ∈Rand eReis a finite-dimensional central division algebra, except when dis an inner derivation induced by an element a in the two-sided Martindale quotient ring of Rsuch that aρp = 0. An analogous result is also proved for the Lie ideal case.     

Derivations with power central values on Lie ideals in prime rings

Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n ₁ ⩾ 0, n ₂ ⩾ 0, n ₃ ⩾ 0, (u ⁿ¹ [d(u), u]u ⁿ²) ⁿ³ ∈ Z(R) for all u ∈ U, then R satisfies S ₄, the standard identity in four variables.     

ON CERTAIN SUBGROUPS OF PRIME RINGS WITH DERIVATIONS

Let R be a noncommutative prime ring and let d be a nonzero derivation on R. A classical theorem of Posner asserts that the subset {[x ^d , x]|x ∈ R} is not contained in the center of R. Under the additional assumption that char R ≠ 2 and d ³ ≠ 0, we show that the additive subgroup of R generated by the subset {[x ^d , x] | x ∈ R} contains a noncentral Lie ideal of R.     

关于素环广义导子和对称双导的几点备注

设R是素环,I是R的非零理想,如果R容许一个非单位映射的左乘子使得对所有x,y∈I满足δ(x°y)=x°y或δ(x°y) x°y=0,那么R可交换.此外,如果R是2-扭自由的素环,U是平方封闭的李理想,γ是伴随导子非零的广义导子,B:R×R→R是迹函数为g(x)=B(x,x)的对称双导,当下列条件之一成立时U为中心李理想(1)γ同态作用于U(2)2[x,y]-g(xy) g(yx)∈Z(R)(3)2[x,y] g(xy)-g(yx)∈Z(R)(4)2(x°y)=g(x)-g(y)(5)2(x°y)=g(y)-g(x)对所有的x,y∈U.     

Commutativity of rings with derivations

I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(x ^m y ⁿ ) = d(y ⁿ x ^m ) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) ^m d(y) ⁿ = d(y) ⁿ d(x) ^m for all x, y ∈ I also implies that R is commutative.     

Annihilators of derivations with Engel conditions on lie ideals

LetR a prime ring,L a noncentral Lie ideal ofR anda ∈R. Suppose thatd is a nonzero derivation ofR such thata[d(u),u] _k =0 for allu ∈L, wherek is a fixed positive integer. Thena=0 except when charR=2 and dim _C RC=4.     

Generalized left derivations acting as homomorphisms and anti-homomorphisms on Lie ideal of rings

Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as a homomorphisms or an anti-homomorphisms on L.     

Skew Derivations and Engel Conditions

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ?)-derivation.     

Derivations on Lie ideals in semiprime rings

We prove that ifR is a semiprime 2-torsion free ring with a derivationd andU a Lie ideal ofR such thatd ² (U)=0, thend(U)?Z(R), the center ofR.     

关于素环平方封闭的李理想的两个结果

Let R be a 2-torsion free prime ring, d1 a nonzero derivation, -γ a generalized derivation associated with a nonzero derivation d2, U a square closed Lie ideal of R. In the present paper,we prove that if [di^2（u）, u] ∈ Z（R） or γ acts as a homomorphism （or an antihomomorphism） on U, then U Z（R）.     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

On Jordan Left Derivations of Lie Ideals in Prime Rings

Let R be a prime ring with characteristic different from two and U be a Lie ideal of R such that u² U for all u U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying d(u²) = 2ud(u), for all u U, then either U Z(R) or d(U) = (0).1991 Mathematics Subject Classification 16W25 16N60     

An identity with generalized derivations on lie ideals, right ideals and Banach algebras

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds:

1. there exists α ε C such that F(x) = α x for all x ε R
2. R satisfies the standard identity s ₄ and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R.

We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.     

On generalized left derivations in rings and Banach algebras

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let \({G: R \longrightarrow R}\) be a generalized Jordan left derivation with associated Jordan left derivation \({\delta: R \longrightarrow R}\). Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer (or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.     

Prime Lie Rings of Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with identity, and δ a nonzero derivation of R such that R is δ-prime. Then Rδ is a prime Lie ring and any nonzero ideal of Rδ contains an ideal of the form Jδ where J is a nonzero δ-ideal of R.     

Certain additive maps on m-power closed Lie ideals

Let R be a prime ring with extended centroid C and m a fixed positive integer >?1. A Lie ideal L of R is called m-power closed if ${u^m \in L}$ for all ${u \in L}$ . We prove that if char R = 0 or a prime p?>?m, then every non-central, m-power closed Lie ideal L of R contains a nonzero ideal of R except when dim _C RC?=?4, m is odd, and ${u^{m-1} \in C}$ for all ${u \in L}$ . Moreover, the additive maps d : L ?? R satisfying d(u ^m )?=?mu ^m-1 d(u) (resp. d(u ^m )?=?u ^m-1 d(u)) for all ${u \in L}$ are completely characterized if char R = 0 or a prime p?>?2(m ? 1).