When is a power of the Frobenius map on a noncommutative ring a homomorphism?

Let φ(x)=xp be the Frobenius map on an associative unital ring R with prime characteristic p. It is well-known that, whenever R is commutative, φn is a ring homomorphism, for all positive integers n. The converse, however, is not true in general. Indeed, we prove that, if R is m-Engel, for some positive integer m, then there exists a positive integer n0 depending only on m such that, for all n≥n0, φn is a ring homomorphism with central image. Conversely, if any one of the following conditions holds: φn respects addition, φn respects multiplication, φn respects Lie multiplication, or the image of φn is commutative, then R is m-Engel, for some m depending only on n. Consequently, if φ is surjective, and any one of the aforementioned conditions holds, then R must be commutative.