# Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$.

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.

Say $A/MA=R\widehat{a}_1+\cdots+R\widehat{a}_n$, where $\widehat{a}_i$ is the residue class of $a_i\in A$ modulo $MA$. Then for $a\in A$ we have $\widehat{a}=r_1\widehat{a}_1+\cdots+r_n\widehat{a}_n$, so $a-(r_1a_1+\cdots+r_na_n)\in MA$. This shows that $A=(Ra_1+\cdots+Ra_n)+MA$, and by NAK we get $A=Ra_1+\cdots+Ra_n$.