Giovanni Scudo

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.

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x 1,…, x n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r 1,…, r n ∈ R, a[F 2(f(r 1,…, r n )), f(r 1,…, r n )] = 0, then one of the following statements hold: 1. a = 0; 2. There exists λ ∈C such that F(x) = λx, for all x ∈ R; 3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c 2 ∈ C; 4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c 2 ∈ C.

Power Values of Generalized Derivations with Annihilator Conditions in Prime Rings

Let R be a prime ring, H a nonzero generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that there exists

Identities with Product of Generalized Skew Derivations on Multilinear Polynomials

{0 \neq a \in R}

Vanishing and cocentralizing generalized derivations on Lie ideals

such that a(usH(u)ut)n = 0 for all

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

{u \in L}