  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 a
centralizer. It is also shown that if $\tau:\T\rightarrow \T$ is an additive
map 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 $ is
an 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 we
obtain some corollaries concerning (Jordan) centralizers and (Jordan)
derivations on triangular rings.
