Minimal number of generators of a module (${\mu }_A(M)={\dim }_{A/\mathfrak{m}}(M/\mathfrak{m}M)(M\text{ finite},A\text{ local})$)

For a module $M$, ${\mu }_A(M)$ is the least cardinality of a generating set, equivalently the least size of a basis of a free module surjecting onto $M$. If $(A,\mathfrak{m})$ is a local ring and $M$ is finitely generated, then ${\mu }_A(M)={\dim }_{A/\mathfrak{m}}(M/\mathfrak{m}M)$. Any generating set spans this quotient, proving the lower bound. Lift a basis of the quotient and let $J$ be the submodule generated by those lifts. Then $N=M/J$ is finitely generated and satisfies $N=\mathfrak{m}N$. For generators $n_i$ write $n_i={\sum }_j{c}_{ij}{n}_j$ with $c_{ij}\in \mathfrak{m}$. The adjugate of $I-C$ shows that $\det (I-C)$ annihilates $N$; this determinant is one modulo $\mathfrak{m}$ and hence a unit. Thus $N=0$, giving the upper bound.