  Let $\Lambda,\Gamma$ be rings and $R=\left(\begin{array}{cc}\Lambda & 0 \\ M
& \Gamma\end{array}\right)$ the triangular matrix ring with $M$ a
$(\Gamma,\Lambda)$-bimodule. Let $X$ be a right $\Lambda$-module and $Y$ a
right $\Gamma$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$
is a silting right $R$-module if and only if both $X_{\Lambda}$ and
$Y_{\Gamma}$ are silting modules and $Y\otimes_\Gamma M$ is generated by $X$.
Furthermore, we prove that if $\Lambda$ and $\Gamma$ are finite dimensional
algebras over an algebraically closed field and $X_{\Lambda}$ and $Y_{\Gamma}$
are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$ is a
support $\tau$-tilting $R$-module if and only if both $X_{\Lambda}$ and
$Y_{\Gamma}$ are support $\tau$-tilting modules, $\Hom_\Lambda(Y\otimes_\Gamma
M,\tau X)=0$ and $\Hom_\Lambda(e\Lambda, Y\otimes_\Gamma M)=0$ with $e$ the
maximal idempotent such that $\Hom_\Lambda(e\Lambda, X)=0$.
