Finitely generated module

An $R$-module is finitely generated when there are finitely many elements $m_1,\dots ,m_r$ such that every element is an $R$-linear combination of them.

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.

Let a finitely generated module over an integral domain have a finite generating set $S$ and a maximal independent subset $T$. For each $s\notin T$, dependence of $T\cup \{s\}$ gives a nonzero coefficient $a_s$ with $a_{s}s\in ⟨T⟩$. The product of those coefficients is nonzero and carries the whole module into $⟨T⟩$.

An R-module is finitely generated exactly when it is the image of a surjection from a finite free module $R^n$. Given generators $m_i$, send $(r_i)$ to ${\sum }_i{r}_i{m}_i$; conversely, the images of the standard free generators generate every such image. This criterion does not assert that the module is free: a quotient can have nontrivial relations.