**Mnemonic:  $\quad M=IM \Rightarrow m=im$**

**The version of Nakayama described:** If $I$ is an arbitrary ideal of an arbitrary ring $A$ and if a finitely generated module $M$ satisfies $M=IM$, then there exists $i\in I$ such that for all $m\in M$ we have $m=im$.       
Please notice: no noetherian nor local assumption on $A$, no assumption at all on $I$.