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Splittable Jordan homomorphisms and commutator ideals

Matej Brešar

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.

Splittable Jordan homomorphisms and commutator ideals

Abstract

We define a Jordan homomorphism from a ring to a ring to be splittable if the ideal (of the subring generated by the image of ) generated by all , , has trivial intersection with the ideal generated by all , . 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.

Paper Structure

This paper contains 4 sections, 17 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Sums and direct sums of homomorphisms and antihomomorphisms
  3. Splittable Jordan homomorphisms
  4. The image of the commutator ideal

Key Result

Lemma 2.7

Suppose a surjective Jordan homomorphism $\varphi:R\to R'$ is the sum of a homomorphism $\varphi_1$ and an antihomomorphism $\varphi_2$ on the ideal $I$. If $\varphi_1(I)\subseteq \varphi(I)$, then $\varphi$ is the direct sum of a homomorphism and an antihomomorphism from the ideal $I$ onto the idea

Theorems & Definitions (43)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Example 2.8
  • Example 2.9
  • ...and 33 more