= Nilpotence index of an ideal
{title2=$\min\{n\ge1:I^n=0\}$}
= Nilpotence indices of an ideal
{synonym}
For a <nilpotent ideal> $I$, its <nilpotence index of an ideal> is the least positive integer $n$ for which $I^n=0$. In $k[X]/(g^e)$, with $g$ an <irreducible polynomial>, the unique <maximal ideal> $(g)$ has index $e$. In a <discrete valuation ring> with $p=u\pi^e$, the <maximal ideal> of its reduction modulo $p$ has the same index. Comparing these indices under a <quotient ring> isomorphism recovers the ramification exponent in the <Kummer-Dedekind theorem>.
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