Relations Between n-Jordan Homomorphisms and n-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(an)=f(a)n , for all a∈A , or f(a1a2⋯an)=f(a1)f(a2)⋯f(an) , for all a1,a2,…,an∈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.