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• Author: Vincenzo De Filippis
Clear All Modify Search  ## Abstract

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R and L is a non-central Lie ideal of R. If [[F(u), u], F(u)] = 0 for all u ∈ L, then either there exists λ ∈ C such that F(x) = λx, for all x ∈ R or R satisfies s4(x1, . . . , x4), the standard identity of degree 4, and there exist a ∈ Qr and λ ∈ C such that F(x) = ax + xa + λx, for all x ∈ R

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## Abstract

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,….,rn) ∈ In. Then one of the following holds:

1. 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 s4, the standard identity of degree 4, and there exist a; a' ∈ U, α,β ∈ C such that G(x) =ax + xa' + αx and H(x) = a'x - xa +βx, for all x ∈ R;

3. R satisfies s4 and there exist a; a' ∈ U, and d : R → R, a derivation of R, such that G(x) = ax + d(x) and H(x)= xa'- d(x), for all x ∈ R, with a + a' ∈ C;

4. R satisfies s4 and there exist a; a' ∈ U, and d : R → R, a derivation of R, such that G(x) = xa + d(x) and H(x) = ax' - d(x), for all x ∈ R, with a - a' ∈ C;

5. there exists e2= e ∈ Soc(RC) such that I = eR and one of the following holds:

(a) [ƒ (x1 ,…., xn); xn + 1] xn+2 is an identity for I;

(b) char (R) = 2 and s4(x1; x2; x3; x4)x5 is an identity for I;

(c) [ƒ (x1 , …, xn)2; xn+1]xn+2 is an identity for I and there exist a, a', b, b' ∈ U,α ∈ C and d : R → R, a derivation of R, such that G(x) = ax + xa' + d(x), H(x)=bx + xb' - d(x), for all x ∈ R, with (a - b' - α) I=(0)=( b-a'-α )I

## Abstract.

Let K be a commutative ring with unity, R be a non-commutative prime K-algebra of characteristic different from 2, U be the Utumi quotient ring of R, C be the extended centroid of R, and G and H be two non-zero generalized derivations of R. Suppose that is a non-central multilinear polynomial over K and there exists such that for all . Then one of the following holds: (1) there exist such that and for all , with ; (2) is central-valued on R and there exist such that and for all , with .

Proceedings of the International Conference held at Aligarh Muslim University, 2016  