n -Jordan Homomorphisms onto Prime Rings

Let [Formula: see text] be a unital ring, [Formula: see text] a prime ring of characteristic greater than [Formula: see text], and [Formula: see text] a surjective [Formula: see text]-Jordan homomorphism. In this paper, we show that [Formula: see text] is either an [Formula: see text]-homomorphism or an anti [Formula: see text]-homomorphism. Furthermore, we prove that [Formula: see text], where [Formula: see text] is either a homomorphism or an anti-homomorphism and [Formula: see text] is an [Formula: see text] root of unity lying in the center of [Formula: see text]. In addition, if [Formula: see text] is a unital Banach algebra and [Formula: see text] is a semi-prime commutative Banach algebra, then each dense range [Formula: see text]-Jordan homomorphism [Formula: see text] is an [Formula: see text]-homomorphism. These results extend Herstein’s theorem to the setting of [Formula: see text]-Jordan homomorphisms.