Let $\T$ be a $2$-torsion free triangular ring and let $\varphi:\T\rightarrow\T$ be an additive map. We prove that if $\A \varphi(\B)+\varphi(\B)\A=0$whenever $\A,\B\in \T$ are such that $\A\B=\B\A=0$, then $\varphi$ is acentralizer. It is also shown that if $\tau:\T\rightarrow \T$ is an additivemap satisfying $\label{t2} X,Y\in \T, \quad XY=YX=0\Rightarrow X\tau(Y)+\delta(X)Y+Y\delta(X)+\tau(Y)X=0$, where $\delta:\T\rightarrow \T $ isan additive map satisfies $X,Y\in \T, \quad XY=YX=0\Rightarrow X\delta(Y)+\delta(X)Y+Y\delta(X)+\delta(Y)X=0$, then $\tau(\A)=d(\A)+\A\tau(\textbf{1})$, where $d:\T\rightarrow \T$ is a derivation and$\tau(\textbf{1})$ lies in the centre of the $\T$. By applying this results weobtain some corollaries concerning (Jordan) centralizers and (Jordan)derivations on triangular rings.
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机译：令$ \ T $为无扭转的$ 2 $三角环，并使$ \ varphi：\ T \ rightarrow \ T $为可加映射。我们证明如果$ \ A \ varphi（\ B）+ \ varphi（\ B）\ A = 0 $每当$ \ A，\ B \ in \ T $都是$ \ A \ B = \ B \ A = 0 $，则$ \ varphi $是集中器。还表明，如果$ \ tau：\ T \ rightarrow \ T $是满足$ \ label {t2} X，Y \ in \ T的加法映射，则\ quad XY = YX = 0 \ Rightarrow X \ tau（Y） + \ delta（X）Y + Y \ delta（X）+ \ tau（Y）X = 0 $，其中$ \ delta：\ T \ rightarrow \ T $是一个附加映射，满足$ X，Y \ in \ T， \ quad XY = YX = 0 \ Rightarrow X \ delta（Y）+ \ delta（X）Y + Y \ delta（X）+ \ delta（Y）X = 0 $，然后$ \ tau（\ A）= d （\ A）+ \ A \ tau（\ textbf {1}）$，其中$ d：\ T \ rightarrow \ T $是一个派生类，$ \ tau（\ textbf {1}）$位于$ \ T $。通过应用该结果，我们获得了有关三角环上的（约旦）扶正器和（约旦）导数的一些推论。
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