Splittable Jordan homomorphisms and commutator ideals

TL;DR

This paper introduces the concept of splittable Jordan homomorphisms and demonstrates that such maps decompose into a sum of a homomorphism and an antihomomorphism on the commutator ideal, providing new structural insights.

Contribution

It defines splittable Jordan homomorphisms and proves they decompose into homomorphisms and antihomomorphisms on the commutator ideal, advancing understanding of Jordan homomorphism structures.

Findings

Splittable Jordan homomorphisms decompose into homomorphisms and antihomomorphisms.

New structural results on Jordan homomorphisms for specific classes of rings.

Enhanced understanding of the ideal structure related to Jordan homomorphisms.

Abstract

We define a Jordan homomorphism $\varphi$ from a ring $R$ to a ring $R'$ to be splittable if the ideal (of the subring generated by the image of $\varphi$) generated by all $\varphi(xy)-\varphi(x)\varphi(y)$, $x,y\in R$, has trivial intersection with the ideal generated by all $\varphi(xy)-\varphi(y)\varphi(x)$, $x,y\in R$. Our main result states that a splittable Jordan homomorphism is the sum of a homomorphism and an antihomomorphism on the commutator ideal. As applications, we obtain results that give new insight into the question of the structure of Jordan homomorphisms on some classes of rings.