---
tag: MM3TQ1WJWF
kind: definition
name: Nilpotent ideal
msc: [16N40]
defines: C2H26BYWHE
uses: [0894J980RK, 5N2CJZ47D4, 0DDA6CY7GJ, NZ1G9WRWFB]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_ideal
---

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $I$ be a [[left ideal|5N2CJZ47D4]], right ideal or [[two-sided ideal|0DDA6CY7GJ]] of $R$.
For $n \ge 1$, let $I^n$ be the additive subgroup of $R$ generated by all products $a_1 a_2 \cdots a_n$ with $a_1, \dots, a_n \in I$.
The ideal $I$ is *nilpotent* if $I^n = 0$ for some $n \ge 1$.

## Remarks

Every nilpotent ideal is a [[nil ideal|NZ1G9WRWFB]], since $I^n = 0$ gives $a^n = 0$ for all $a \in I$.
Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $K$ and let $I$ be the set of its matrices with zero diagonal.
$I$ is a two-sided ideal, and the product of two matrices of the form $\begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix}$ is zero, so $I^2 = 0$.
Non-example: let $R = \Z[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots)$ and let $I$ be the ideal generated by the images of all $x_k$.
$I$ is nil, since it is generated by nilpotent elements of a commutative ring.
It is not nilpotent, because for every $n$ the product $x_{n+1}^{n}$ of $n$ elements of $I$ is nonzero in $R$.