Faithful modules and integral extensions

Question

Suppose $R\subseteq S$ is an integral extension, $S$ is a finite $R$-module. Is it true that $S$ is faithful $R$ module? (To show faithfulness we need to show $\operatorname{ann}_R(S)=0$).

Answer 1

If $R$ is a unital ring, and a subring of $S$, then the $R$-annhilator of $S$ is zero, since only $0$ annihilates $1\in S$.