Power Central Values Research Articles

Generalized derivations with powercentral values on Lie ideals and Banach Algebras

Generalized derivations with powercentral values on Lie ideals and Banach Algebras

Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Power-Central Values and Engel Conditions in Prime Rings with Generalized Skew Derivations

Let R be a prime ring of characteristic different from 2 with extended centroid C, $$n\ge 1$$ a fixed positive integer, $$F, G:R\rightarrow R$$ two non-zero generalized skew derivations of R.

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

Abstract Let ℜ {\mathfrak{R}} be a ring with center Z ⁢ ( ℜ ) {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 𝔉 {\mathfrak{F}} such that ( [ 𝔉 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every s ∈ ℜ {s\in\mathfrak{R}} , then either 𝔉 ⁢ ( s ) = p ⁢ s {\mathfrak{F}(s)=ps} for every s ∈ ℜ {s\in\mathfrak{R}} or ℜ {\mathfrak{R}} satisfies s 4 {s_{4}} and 𝔉 ⁢ ( s ) = s ⁢ p {\mathfrak{F}(s)=sp} for every s ∈ ℜ {s\in\mathfrak{R}} and p ∈ 𝔘 {p\in\mathfrak{U}} , the Utumi quotient ring of ℜ {\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.

Generalized derivations with nilpotent, power-central, and invertible values in prime and semiprime rings

Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R, and fixed integers. Let be a non-zero multilinear polynomial over C in t non-commuting variables, be any subset of R and . We prove the following results:If R is prime and for all , then is central valued on R.If R is prime and , for all , then is power central valued on R, unless .If R is semiprime and for all , then , for any and , that is there exists a central idempotent element such that , d vanishes identically on eQ and is central valued on .If R is semiprime and is zero or invertible in R, for all , then either R is a division ring or it is the ring of 2 × 2 matrices over a division ring, unless when , for any and .If R is prime and I is a non-zero right ideal of R such that and , for all , then is an identity on I.Let R be prime and I a non-zero right ideal of R such that and , for all . If there exists such that , then either is power central valued on R or is an identity on I, unless .

Skew Derivations with Power Central Values on Commutators

Let R be a prime ring with center [Formula: see text], δ: R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if (i) [δ([x,y]),[x,y]]n = 0 for all x, y ∈ R or (ii) [Formula: see text] for all x ∈ R, except some specific cases.

Generalized Skew Derivations with Power Central Values on Lie Ideals

Let R be a prime ring with center Z and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f(x) n ∈ Z for all x ∈ L, where n is a fixed positive integer. Then f = 0 unless dim C RC = 4.

Identities with Engel conditions on derivations

Let R be a semiprime ring with a derivation D. The focus is on the two identities with Engel condition on $${D: [x^m, D(x^{n_1}),\ldots,D(x^{n_s})]_s=0}$$ for all $${x\in R}$$ and $${[x^m, D(x)^{n_1},\ldots,D(x)^{n_s}]_s=0}$$ for all $${x\in R}$$ , where s, m, n 1, . . . , n s are fixed positive integers. Our results are natural generalizations of Posner’s theorem on centralizing derivations, Herstein’s theorem on derivations with power-central values and a recent result by A. Fošner, M. Fošner and Vukman.

Generalized Derivations with Power Central Values on Multilinear Polynomials on Right Ideals

Generalized Derivations with Power Central Values on Multilinear Polynomials on Right Ideals

Derivations with power central values on multilinear polynomials in prime rings

Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x1, ..., xn) a nonzero multilinear polynomial over C. Suppose that xsd(x)xt ∈ Z(R) for all x ∈ {d(f(x1, ..., xn))|x1, ..., xn ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρC = eRC for some idempotent e in the socle of RC and f(x1, ..., xn)N is central-valued in eRCe, where N = s + t + 1.

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.

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.

Generalized Derivations with Power-Central Values on Multilinear Polynomials

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.

Commutators with power central values on a Lie ideal

Commutators with power central values on a Lie ideal

A note on derivations with power central values on a Lie ideal

Let R be a prime ring of characteristi c Φ 2 with a derivation d Φ 0 and a non-central Lie ideal U such that d(u)n is central, for all u e U. We prove that R must satisfy s4, the standard identity in 4 variables; hence R is either commutative or an order in a 4-diniensional simple algebra. This result extends a theorem of Herstein to Lie ideals. In [2] Herstein shows that if R is a prime ring with center Z and if d Φ 0 is a derivation of R such that J(JC) W e Z, for all x e 7?, then i? satisfies s49 the standard identity in 4 variables. This theorem indicates that the global structure of a ring is often tightly connected to the behaviour of one of its derivations. The purpose of this note is to show that the same conclusion holds for prime rings of characteristi c Φ 2 even if we assume only that d(u)n is central for those u in some non-central Lie ideal. We will proceed by first proving the result when d is inner. We will then use Kharchenko's theorem on differential identities [5] to reduce to the case where d is inner on the Martindale quotient ring of R. By using Kharchenko's theorem, our proof will actually be somewhat simpler than the proof in [2]. In all that follows, unless stated otherwise, R will be a prime ring of characteristic Φ 2, U a non-central Lie ideal of i?, d Φ 0 a derivation of /?, and n > 1 a fixed integer such that d(u)n is central, for all u e U. For any ring S, Z = Z(S) will denote its center. For subsets A, B c R9 [A, B] will be the additive subgroup generated by all [α, b] = αb — bα; α e A, b e B. In addition, s4 will denote the standard identity in 4 variables. By a result of Herstein [3], U D [/, R] for some / Φ 0, an ideal of R. Therefore, we will assume throughout that U D [/, R]. We will also make frequent and important use of the following three results. We do not necessarily state them in their fullest generality. 1. (Carini-Giamb runo [1].) If U <£ Z(R) is a Lie ideal of a prime ring R of characteristic Φ 2, and if d is a derivation of R such that d(u)n = 0, for all WG [/, then d = 0.