A note on $n$-Jordan homomorphisms

TL;DR

This paper explores the relationship between Jordan homomorphisms and n-Jordan homomorphisms in ring theory, extending a theorem to show conditions under which n-Jordan homomorphisms are actually n-homomorphisms.

Contribution

It generalizes G. An's result by using a variation of Herstein's theorem to establish when n-Jordan homomorphisms coincide with n-homomorphisms in rings.

Findings

Under certain conditions, n-Jordan homomorphisms are equivalent to n-homomorphisms.

The result applies to rings with a unit and characteristic conditions on the codomain.

The paper extends previous theorems in the structure theory of ring homomorphisms.

Abstract

By using a variation of a theorem on $n$-Jordan homomorphisms due to Herstein, we deduce the following G. An's result: Let $ A $ and $ B $ be two rings where $ A $ has a unit and $ char(B)> n. $ If every Jordan homomorphism from $ A $ into $ B $ is a homomorphism (anti-homomorphism), then every $n$-Jordan homomorphism from $ A $ into $ B $ is an $n$-homomorphism (anti-$n$-homomorphism).