If every Jordan homomorphism from ring A to ring B is a homomorphism or anti-homomorphism, then every n-Jordan homomorphism from A to B is an n-homomorphism or anti-n-homomorphism, provided A has a unit and char(B) > n.
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A note on $n$-Jordan homomorphisms
If every Jordan homomorphism from ring A to ring B is a homomorphism or anti-homomorphism, then every n-Jordan homomorphism from A to B is an n-homomorphism or anti-n-homomorphism, provided A has a unit and char(B) > n.