modules which have faithful sub-modules

Question

An $R$-Module $M$ is faithful if $\{a \in R \mid\forall m\in M \space \space am=0\}=\{0\} \space($ i.e the annihilator of $M$ in $R$ is zero). Is there a module $M$ which has a faithful submodule $N$ but $M$ not be faithful?

Answer 1

If $M$ is not faithful, then there is $a\in R$ with $a\neq 0$ such that $am=0$ for every $m\in M$. Since every submodule $N$ of $M$ is also a subset of $M$, this means that for every $n\in N$ $an=0$. Thus, $N$ cannot be faithful.