$R$ be a commutative ring with $1$ and maximal ideal $m$. Let $f:M \rightarrow N$ be an $R$-linear map and $N$ is a finitely generated $R$-module. Suppose that the induced homomorphism $M\otimes_R(R/m) \rightarrow N\otimes(R/m)$ is surjective. Now if $(R,m)$ is local ring then show that $f$ is surjective.
What I obtained so far as following:
$N$ is f.g. so is $N/mN$. Assume $N/mN=\left<x_1+mN,x_2+mN,\dots,x_n+mN\right>$. Since $(R,m)$ is local by NAK we have $N=\left<x_1,x_2,\dots,x_n\right>$. Now we know $M/mM \cong M\otimes_R(R/m)$ via $\overline{m}\longmapsto m\otimes\overline{1}$ and similarly for $N/mN$. So we have a surjection from $M/mM \rightarrow N/mN $ given by $\overline{m} \longmapsto \overline{f(m)}$. So there are $u_1,u_2,\dots,u_n\in M$ such that $\overline {f(u_i)}=\overline {x_i}$. But I can't show that there is $u_i$ such that $f(u_i)=x_i$. Help me. Thanks.
Apply NAK to $\mathrm{coker}(f)$.