= Generator bound for primary-ideal length
{title2=$\operatorname{length}(R/I^t)\leq L\binom{t+n-1}{n}$}
If $I$ is an $\mathfrak m$-primary <ideal> generated by $n>0$ elements in a <Noetherian local ring> and $L=\operatorname{length}(R/I)$, then $I^j/I^{j+1}$ is a quotient of $\binom{j+n-1}{n-1}$ copies of $R/I$. Summing their lengths gives the displayed bound. Since $I\subseteq\mathfrak m$, it also bounds $\operatorname{length}(R/\mathfrak m^t)$. Together with the <prime-chain lower bound for local length>, localization at a minimal prime proves the <Krull height theorem>.
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