Is annihilator of maximal submodule is a maximal ideal?

Question

Let $R$ be a commutative ring with identity element and $M$ is an $R$-module. We know the annihilator of a submodule $N$ of $M$ ($I=(N:M)$) is an ideal in $R$. If $N$ is a maximal submodule of $M$, is ideal $I$ (annihilator of $N$ in $M$) maximal in $R$?

Answer 1

Indeed $M/N$ is a simple module, i.e. has no nontrivial submodules; hence $M/N =Rm$ for any $m \neq 0$ in $M/N$. Hence, as an $R$-module, $M/N$ is isomorphic to $R/I$ where $I = (N:M)$. But an $R$-module of the form $R/I$ is a simple module if and only if $I$ is maximal.