Nilpotent ideal

I

is nilpotent when

I^{r} = 0

for some positive integer

r

.

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- Nilpotence index of an ideal Nilpotent ideal

Nilpotence index of an ideal ( $min {n \geq 1 : I^{n} = 0}$ )

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 π^{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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Nilpotent ideal by

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In abstract algebra, specifically in the study of rings, a **nilpotent ideal** is an ideal such that there exists some positive integer \( n \) for which the \( n \)-th power of the ideal is equal to the zero ideal.

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