On finite generation of $IM$ where $M$ is an $R$-module and $I=Ann_R (M/N)$ , where $N$ is a finitely generated submodule of $M$

Question

Let $M$ be an $R$-module , where $R$ is a commutative ring with unity. Let $N$ be a finitely generated submodule of $M$ and let $I=Ann_R (M/N)$ . Then is $IM$ a finitely generated submodule of $M$ ?

Answer 1

Let $R$ be the polynomial ring over a field in infinitely many variables $x_i, i\in\mathbb{N}$. Let $M$ be the maximal ideal generated by all the $x_i$s and let $N=x_1R\subset M$. Then $\mathrm{Ann}_R(M/N)=x_1R$, but $x_1M$ is not finitely generated.