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

Question

Let $M$ be a finitely generated $R$-module, where $R$ is a commutative ring with unity. Let $N$ be a submodule of $R$ such that $IM$ is finitely generated, where $I=\operatorname{Ann}_R (M/N)$ . Then, is $N$ finitely generated ?