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 $x\in A$.

令A和B为单位环。附加地图$吨:A\to 乙$被称为加权约旦同态如果$C=吨(\mathrm{1个})$是一个可逆的中心元素，并且$C吨(X^{\mathrm{2个}})=吨(X{)}^{\mathrm{2个}}$对全部$X\in A$. 我们提供假设，这些假设在以下情况下特别得到满足$A=乙={米}_n(R)$和$n\ge \mathrm{2个}$和R任何单位环$\frac{\mathrm{1个}}{\mathrm{2个}}$, 在其下每个满射加法映射$吨:A\to 乙$与财产$吨(X)吨(是)+吨(是)吨(X)=0$只要xy  =  yx  = 0 是加权约旦同态。此外，我们证明如果A是具有 char 的素环$(A)\ne \mathrm{2个},\mathrm{3个},\mathrm{5个}$, 那么一个双射加法映射$吨:A\to A$是加权 Jordan 同态，前提是存在加性映射$\mathrm{小号}:A\to A$这样$\mathrm{小号}(X^{\mathrm{2个}})=吨(X{)}^{\mathrm{2个}}$对全部$X\in A$.