= Primary ideals can need more generators than the maximal ideal
{title2=$\mu(\mathfrak m^r)=r+1\text{ in }k[[u,v]]$}
In the <Noetherian local ring> $A=k[[u,v]]$ with <maximal ideal> $\mathfrak m=(u,v)$, the <embedding dimension> is two but $\mathfrak m^r$ needs $r+1$ generators for every $r\geq1$. The degree-$r$ monomials $u^r,u^{r-1}v,\ldots,v^r$ generate $\mathfrak m^r$, and their images form a <basis> of $\mathfrak m^r/\mathfrak m^{r+1}$, so no smaller generating list exists. These <ideals> are $\mathfrak m$-primary: if $ab\in\mathfrak m^r$ and $a\notin\mathfrak m^r$, then $b$ cannot be a <unit>; hence $b\in\mathfrak m$ and $b^r\in\mathfrak m^r$. Consequently <embedding dimension> bounds the minimum over all maximal-primary <ideals>, because the <maximal ideal> itself is eligible, but does not bound the number of generators of each such <ideal>.
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