n-Jordan Homomorphisms into Fields and Integral Domains

We investigate under what conditions an n-Jordan homomorphism between rings or algebras is an n-homomorphism. More specifically, if R and S are rings and S is of characteristic greater than n, along with some other results for an n-Jordan homomorphism ψ : R → S, we prove the following results: If S is a field and either char S = 0 or char S 2 card {α ∈ S : α^n−1 = 1}, then ψ is an n-homomorphism, where char S and card denote the characteristic of S and the cardinal number (of a set), respectively. If S is an integral domain, then ψ is an n-homomorphism if S is an algebra, or either one of the conditions char S = 0, char S 2(n − 1), n = 3, holds.