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We investigate under what conditions  n -Jordan homomorphisms between rings are  n -homomorphism, or homomorphism; and under what conditions,  n -Jordan homomorphisms are continuous. One of the main goals in this work is to show that every  n -Jordan homomorphism  f  :  A → B , from a unital ring  A  into a ring  B  with characteristic greater than  n , is a multiple of a Jordan homomorphism and hence, it is an  n -homomorphism if every Jordan homomorphism from  A  into  B  is a homomorphism. In particular, if  B  is an integral domain whose characteristic is greater than  n , then every  n -Jordan homomorphism  f  :  A → B  is an  n -homomorphism. Along with some other results, we show that if  A  and  B  are unital rings such that the characteristic of  B  is greater than  n , then every unital  n -Jordan homomorphism  f  :  A → B  is a Jordan homomorphism and hence, it is an  m -Jordan homomorphism for any positive integer  m ≥  2. We also investigate the automatic continuity of  n -Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the  n -Jordan homomorphism has dense range.

Certain properties of Jordan homomorphisms, n -Jordan homomorphisms and n -homomorphisms on rings and Banach algebras