An engel condition with automorphisms for left ideals

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[...[[σ(x ⁿ⁰), xⁿ¹], xⁿ²], ...], x^nk] = 0 for all x ∈ L, where n₀, n₁, n₂, ..., n _k are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.