= Maximal-primary ideal criterion
{title2=$\mathfrak m^n\subseteq Q\subseteq\mathfrak m$}
In a <Noetherian ring>, an <ideal> $Q$ is $\mathfrak m$-<primary> for a <maximal ideal> $\mathfrak m$ if and only if $\mathfrak m^n\subseteq Q\subseteq\mathfrak m$ for some $n\geq1$. For necessity, choose generators $x_1,\ldots,x_d$ of $\mathfrak m$ and exponents $e_i\geq1$ with $x_i^{e_i}\in Q$. Every product of $1+\sum_i(e_i-1)$ generators lies in $Q$. If $\mathfrak m=0$, take $n=1$. Conversely the inclusions imply $\sqrt Q=\mathfrak m$, so $R/Q$ is a <local ring> with <maximal ideal> $\mathfrak m/Q$. Every element outside this <maximal ideal> is a <unit>, while every element inside is <nilpotent>. Therefore every <zero divisor> is <nilpotent>, which is the <primary ideal> condition. The reverse implication works for any <commutative ring>.
\Comment[[
Exact target: primary-ideals-for-a-regular-principal-prime
Existing parent: Primary ideal
Intended file: /home/ciro/bak/git/codex-wiki/commutative-algebra.bigb
]]
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