In the Wikipedia there is an application of Nakayama lemma:
In the special case of a finitely generated module $M$ over a local ring $R$ with maximal ideal $m$, the quotient $M/mM$ is a vector space over the field $R/m$.
My confusion is why we need the ring to be local, and how is the operation defined for the scalar multiplication (M/mM over R/m)? And how is this related to Nakayama lemma?
Define $R/m\times M/mM\to M/mM$ by $(\bar a,\hat x)\mapsto\widehat{ax}$. This is well defined: if $\bar a=\bar a'$, and $\hat x=\hat x'$, then $a-a'\in m$ and $x-x'\in mM$, so $ax-a'x'=a(x-x')+(a-a')x'\in mM$, so $\widehat{ax}=\widehat{a'x'}$. (This definition has nothing to do with $M$ finitely generated or $R$ local. The ideal $m$ can be also arbitrary.)
There it is also said: "a basis of $M/mM$ lifts to a minimal set of generators of $M$". This happens for if $\hat x_1,\dots,\hat x_n$ is an $R/m$-basis for $M/mM$, then $M/N=m(M/N)$, where $N=\langle x_1,\dots,x_n\rangle$, so by NAK we have $M/N=0$, that is, $M=N$.