= Minimal number of generators
{title2=$\mu_R(M)$}
{wiki}
The <minimal number of generators> of a <group>, <module>, <ideal>, or <algebraic structure> is the smallest size of a <generating set> for the specified operations. The type of structure matters: generating a <ring> as an <ideal>, as an <algebra>, and as an additive <group> are different questions.
For a finitely generated <module> $M$ over a <local ring> $(R,\mathfrak m)$ with <residue field> $k$, <Nakayama's lemma> gives $\mu_R(M)=\dim_k(M/\mathfrak mM)$. Every <generating set> maps to a spanning set, giving the <lower bound>; lifts of a <basis> generate by <Nakayama's lemma>, giving equality. For example, the <ideal> $(x,y)$ in $k[x,y]_{(x,y)}$ needs two generators, since its quotient by $(x,y)^2$ has <basis> given by the classes of $x,y$.
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