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 $I$.

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

The version of Nakayama described: If $I$ is an arbitrary ideal of an arbitrary commutative 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 $I$.