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 (x^{n) =h (x) ^n for all x A; h is called an n-homomorphism or an anti-n-homomorphism if h (₈=₁^nx₈) =₈=₁^n h (x₈) or h (₈=₁^nx₈) =₈=₀^n-1 h (x₍-₈), respectively, for all x₁,. . . , x₍ 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), where h (e) is in the centralizer of h (A) and 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.