Weighted Jordan homomorphisms

TL;DR

This paper characterizes weighted Jordan homomorphisms in unital rings, especially matrix rings, by identifying conditions under which additive maps preserving certain zero products are such homomorphisms.

Contribution

It establishes new conditions under which additive maps are weighted Jordan homomorphisms, extending known results to broader classes of rings including matrix and prime rings.

Findings

Surjective maps preserving zero products are weighted Jordan homomorphisms under certain conditions.

Bijective maps on prime rings are weighted Jordan homomorphisms if related to a map satisfying a quadratic condition.

Results apply to matrix rings over rings with 1/2, generalizing previous characterizations.

Abstract

Let $A$ and $B$ be unital rings. An additive map $T:A\to B$ is called a weighted Jordan homomorphism if $c=T(1)$ is an invertible central element and $cT(x^2) = T(x)^2$ for all $x\in A$. We provide assumptions, which are in particular fulfilled when $A=B=M_n(R)$ with $n\ge 2$ and $R$ any unital ring with $\frac{1}{2}$, under which every surjective additive map $T:A\to B$ with the property that $T(x)T(y)+T(y)T(x)=0$ whenever $xy=yx=0$ is a weighted Jordan homomorphism. Further, we show that if $A$ is a prime ring with char$(A)\ne 2,3,5$, then a bijective additive map $T:A\to A$ is a weighted Jordan homomorphism provided that there exists an additive map $S:A\to A$ such that $S(x^2)=T(x)^2$ for all $x\in A$.