Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module s.t. $M=IM$. How to prove: $$\exists a \in I \text{ such that } (1-a)M=0. $$
I tried to solve: $\exists a \in I $ s.t. $(1-a)M=0 $ i.e. $M=IM=aM$ i.e $M$ is cycle.$$(1M-aM=0 \rightarrow 1M=aM \rightarrow M=aM)$$
Hint. Use your previous question with $\phi=1_M$.