A note on n-Jordan homomorphisms

{Let $ A, B $ be two rings and n 2 be an integer. An additive map h A→ B is called an n-Jordan homomorphism if h(xⁿ)=h(x)ⁿ for all x∈ A; h is called an n-homomorphism or an anti-n-homomorphism if h(∏ᵢ₌₁ⁿxᵢ)=∏ᵢ₌₁ⁿ h(xᵢ) or h(∏ᵢ₌₁ⁿxᵢ)=∏ᵢ₌₀ⁿ⁻¹ h(xₙ₋ᵢ), respectively, for all x₁,...,xₙ∈ A.} {We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let n≥ 2 be an integer and h be an $n-$Jordan homomorphism from a ring A into a ring B of characteristic greater than n. Suppose further that A has a unit e, then h = h(e)τ, where $h(e)$ is in the centralizer of $h(A)$ and τ is a Jordan homomorphism.} {By using this variation, we deduce the following result of G. An: Let A and Bbe two rings, where A has a unit and B is of characteristic greater than an integer n ≥ 2. If every Jordan homomorphism from A into B is a homomorphism (anti-homomorphism), then every $n-$Jordan homomorphism from A into B is an n-homomorphism (anti-n-homomorphism).As a consequence of an appropriate lemma, we also obtain the following resultof E. Gselmann: Let $A, B$ be two commutative rings and B is of characteristic greater than an integer n≥ 2. Then every n-Jordan homomorphism from A intoB is an $n-$homomorphism.}