Relations Betweenn-Jordan Homomorphisms andn-Homomorphisms

For n >= 2, an additive map f between two rings A and B is called an n-Jordan homomorphism, or an n-homomorphism if f (a(n)) = f (a)(n), for all a is an element of A, or f (a(1)a(2) ... a(n)) = f (a(1)) f (a(2)) ... f (a(n)), for all a(1), a(2),..., a(n) is an element of A, respectively. In particular, if n = 2 then f is simply called a Jordan homomorphism or a homomorphism, respectively. The notion of n-Jordan homomorphism between rings was introduced in 1956 by Herstein and the concept of n-homomorphism between algebras was introduced in 2005 by Hejazian et al. Properties of n-Jordan homomorphisms as well as n-homomorphisms have been studied by many authors since then. One of the main questions is that, "under what conditions n- Jordan homomorphisms are n-homomorphism?". Another natural question is that "under what conditions certain properties of homomorphisms may be extended to n-homomorphisms". We provide conditions under which these questions have affirmative answers. We also study the continuity problem for n-Jordan homomorphisms on Banach algebras, while extending some known results in this field.