Power Values of Generalized Derivations with Annihilator Conditions in Prime Rings

Abstract

Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R, and F a generalized derivation with associated derivation d of R, m ≥ 1, n ≥ 1 two fixed integers, and 0 ≠ a ∈ R.

1. Assume that a((F([x, y]))^m − [x, y]ⁿ) = 0, for all x, y ∈ I, then one of the following statements holds:

   1. R is commutative;

   2. n = m = 1 and there exists b ∈ Q such that F(x) = bx for all x ∈ R with ab = a;

   3. There exists b ∈ C such that F(x) = bx for all x ∈ R with b^m = 1 and [x, y]^m = [x, y]ⁿ, for all x, y ∈ R;

   4. R ⊆ M₂(C), the ring of 2 × 2 matrices over C; n = 1 and m ≥ 2 such that α^m = α for all α ∈C; and there exists b ∈ Q such that F(x) = bx for all x ∈ R with ab = a;

   5. R ⊆ M₂(C) and char(R) = 2.

2. Assume that char(R) ≠ 2 and a((F([x, y]))^m − [x, y]ⁿ) ∈ Z(R) for all x, y ∈ I. If there exist x₀, y₀ ∈ I such that a((F([x₀, y₀]))^m − [x₀, y₀]ⁿ) ≠ 0, then either there exists a field E such that R ⊆ M₂(E) or a ∈ Z(R), [x, y]^m − [x, y]ⁿ ∈ Z(R) for any x, y ∈ R, and there exist b, c ∈ Z(R) sucht that (b + c)^m = 1.

Key Words:

• Extended centroid
• Generalized derivation
• Prime ring
• Utumi quotient ring

2010 Mathematics Subject Classification:

• 16N60
• 16U80
• 16W25