For \(n\ge 2\), 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 \(a\in A\), or \(f(a_1a_2\cdots a_n)=f(a_1)f(a_2)\cdots f(a_n)\), for all \(a_1,a_2,\ldots ,a_n\in A\), 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.

对于\（N \ GE 2 \） ，添加剂地图˚F两个环之间甲和乙称为Ñ -Jordan同态，或Ñ -homomorphism如果\（F（A ^ N）= F（A）^ N \ ），对于所有\（一个在A \ \） ，或\（F（a_1a_2 \ cdots A_N）= F（A_1）F（A_2）\ cdots F（A_N）\） ，对于所有\（A_1，A_2，\ ldots，a_n \在A \中）。特别地，如果\（n = 2 \），则f分别简单地称为约旦同构或同构。环之间的n-约旦同态概念是1956年由Herstein和n概念引入的Hejazian等人在2005年引入了代数之间的同态。性能ñ -Jordan同态以及ñ从那时起-homomorphisms已经研究了许多作者。主要问题之一是，“在什么条件下，n -Jordan同态是n-同态？”。另一个自然的问题是“在什么条件下，同态的某些性质可以扩展到n-同态”。我们提供了这些问题得到肯定答案的条件。我们还研究了Banach代数上n -Jordan同构的连续性问题，同时扩展了该领域的一些已知结果。