Annihilators of derivations with Engel conditions on one-sided ideals

Abstract

Let R be a noncommutative prime ring with extended centroid C, two-sided Martindale quotient ring Q and λ a nonzero left ideal of R. Suppose that D is a nonzero derivation of R and 0where k and n are fixed positive integers. Then D = ad (b) for some b ∈ Q such that λb = 0 and ab = 0. We also prove an analogous result for right ideals.

Annihilators of skew derivations with Engel conditions on prime rings

“…And by Theorem A, we obtain that either a = 0 or c ∈ C. By assumption we conclude c ∈ C and a[y n , x n ] k = 0 for all x ∈ Q. Then by the proof of [25], Proposition 3, we obtain that R is commutative, a contradiction.…”

Derivations in commutators with power central values in rings

“…A significant extension of [6] shows that R is commutative if [d(x k ), x k ] n = 0 for all x in a nonzero left ideal of R (see [11,Theorem 1]). In [15] Shiue discussed the situation when a[d(x k ), x k ] n = 0 for all x in an one-sided ideal of R, where 0 = a ∈ R.…”