Let $R$ be a local commutative ring with the maximal ideal $M$. Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$.
I tried to apply Nakayama's lemma but I don't know what should we do in the case that $MA\neq M$ because the ring is not a $PID$. Thanks for any help.