Vincenzo De Filippis

A product of two generalized derivations on polynomials in prime rings

LetR be a prime ring of characteristic different from 2,U the Utumi quotient ring ofR, C the extended centroid ofR, F andG non-zero generalized derivations ofR andf(x1, ...,xn) a polynomial overC. Denote byf(R) the set {f(r1, ..., rn): r1, ..., rn ∃ R} of all the evaluations off(x1, ...,xn) inR. Suppose thatf(x1, ...,xn) is not central valued onR. IfR does not embed inM2(K), the algebra of 2 × 2 matrices over a fieldK, and the composition (FG) acts as a generalized derivation on the elements off(R), then (FG) is a generalized derivation of R and one of the following holds:1.there existsα ∈ C such thatF(x)=αx, for allx ∈ R;2.there existsα ∈ C such thatG(x)=αx, for allx ∈ R;3.there exista; b ∈ U such thatF(x)=ax, G(x)=bx, for allx ∈ R;4.there exista; b ∈ U such thatF(x)=xa, G(x)=xb, for allx ∈ R;5.there exista; b ∈ U, α,β ∈ C such thatF(x)=ax+xb, G(x)=αx+β(αx − xb), for allx ∈ R.

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r k ), r k ] n = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c − α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.

Strong commutativity and Engel condition preserving maps in prime and semiprime rings

Let ℛ be a prime ring of characteristic different from 2, 𝒰 its right Ututmi quotient ring, 𝒞 its extended centroid, f(x 1, … , x n ) a multilinear polynomial in n non-commuting variables over 𝒞 and S = { f(r 1, … , r n ) : r 1, … , r n  ∈ ℛ}. Let F: ℛ → ℛ and G: ℛ → ℛ be non-zero generalized derivations on ℛ. We say that F and G are mutually strong Engel condition preserving (SEP for brevity) on 𝒮 if [G(x), F(y)] h  = [x, y] h , for all x, y ∈ 𝒮 and fixed h ≥ 1. In this article we show that, if f(x 1, … , x n ) is not central valued on ℛ and F, G are mutually SEP on 𝒮, then one of the following holds: (a) there exists λ ∈ 𝒞 such that, for any x ∈ ℛ, G(x) = λx and F(x) = λ−h x; (b) char(R) = p ≥ 3 and there exist λ ∈ 𝒞 and s ≥ 1 such that, for any x ∈ ℛ, G(x) = λx and is central valued on ℛ; (c) ℛ satisfies s 4, the standard identity of degree 4. The semiprime case for mutually SEP derivations on Lie ideals is also considered.

GENERALIZED DERIVATIONS IN PRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS

Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and δ a generalized derivations of R. If [[δ(x), x], δ(x)] = 0 for all x ∈ R then either R is commutative or δ(x) = ax for all x ∈ R and some a ∈ C. We also obtain some related result in case R is a Banach algebra and δ is either continuous or spectrally

Vanishing Derivations and Centralizers of Generalized Derivations on Multilinear Polynomials

Let R be a prime algebra over a commutative ring K with unity, and let f(x 1,…, x n ) be a multilinear polynomial over K, not central valued on R. Suppose that d is a nonzero derivation of R and G is a nonzero generalized derivation of R such that for all r 1,…, r n ∈ R. If the characteristic of R is different from 2, then one of the following holds: 1. There exists λ ∈C, the extended centroid of R, such that G(x) = λx, for all x ∈ R; 2. There exist a ∈ U, the Utumi quotient ring of R, and λ ∈C = Z(U) such that G(x) = ax + xa + λx, for all x ∈ R, and f(x 1,…, x n )2 is central valued on R

On the annihilator of commutators with derivation in prime rings

SianoR un anello primo di caratteristica differente da 2,d una derivazione non nulla diR, L un ideale di Lie non centrale diR, a ∈ R. Sea[d(u), u] = 0, per ogni scelta diu ∈ L, alloraa = 0.

NOTES ON GENERALIZED DERIVATIONS ON LIE IDEALS IN PRIME RINGS

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that u s H(u)u t = 0 for all u 2 L, where s ‚ 0,t ‚ 0 are fixed integers. Then H(x) = 0 for all x 2 R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x,y 2 R, the commutator xyiyx will be denoted by (x,y). An additive mapping d from R to R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x,y 2 R. A derivation d is inner if there exists a 2 R such that d(x) = (a,x) holds for all x 2 R. An additive subgroup L of R is said to be a Lie ideal of R if (u,r) 2 L for all u 2 L, r 2 R. The Lie ideal L is said to be noncommutative if (L,L) 6 0. Hvala (8) introduced the notion of generalized derivation in rings. An additive mapping H from R to R is called a generalized derivation if there exists a derivation d from R to R such that H(xy) = H(x)y+xd(y) holds for all x,y 2 R. Thus the generalized derivation covers both the concepts of derivation and left multiplier mapping. The left multiplier mapping means an additive mapping F from R to R satisfying F(xy) = F(x)y for all x,y 2 R. Throughout this paper R will always present a prime ring with center Z(R), extended centroid C and U its Utumi quotient ring. It is well known that if ‰ is a right ideal of R such that u n = 0 for all u 2 ‰, where n is a fixed positive integer, then ‰ = 0 (7, Lemma 1.1). In (2), Chang and Lin consider the situation when d(u)u n = 0 for all u 2 ‰ and u n d(u) = 0 for all u 2 ‰, where ‰ is a nonzero right ideal of R. More precisely, they proved the following: Let R be a prime ring, ‰ a nonzero right ideal of R, d a derivation of R and n a fixed positive integer. If d(u)u n = 0 for all u 2 ‰, then d(‰)‰ = 0 and if u n d(u) = 0 for all u 2 ‰, then d = 0 unless R » M2(F), the 2◊2 matrices over a field F of two elements.

GENERALIZED DERIVATIONS ON SEMIPRIME RINGS

Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, . Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and for all x, , then either R is commutative or n = 1, , R contains a non-zero central ideal and for all .

Posner's second theorem, multilinear polynomials and vanishing derivations

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d and δ non-zero derivations of R , f (x 1 ,…, x n ) a multilinear polynomial over K .If then f ( x 1 ,…, x n is central-valued on R.

Commuting Values of Generalized Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x 1,…, x n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x 1,…, x n ) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds: 1. f(x 1,…, x n )2 is central valued on R; 2. R satisfies s 4, the standard identity of degree 4.