  In this paper, we study Serre's condition $(S_n)$ for tensor products of
modules over a commutative noetherian local ring. The paper aims to show the
following. Let $M$ and $N$ be finitely generated module over a commutative
noetherian local ring $R$, either of which is $(n+1)$-Tor-rigid. If the tensor
product $M \otimes_R N$ satisfies $(S_{n+1})$, then under some assumptions
$\mathrm{Tor}_{i}^R(M, N) = 0$ for all $i \ge 1$. The key role is played by
$(n+1)$-Tor-rigidity of modules. As applications, we will show that the result
recovers several known results.
