Nilpotent element

Definition

Let $R$ be a ring. An element $x \in R$ is nilpotent if there is a positive integer $n$ such that $x^n = 0$.

Remarks

In $\mathbb{Z}/8\mathbb{Z}$, the class of $2$ is nilpotent, because $2^3 = 8 = 0$. The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent: its square is the zero matrix. In a nonzero ring, a nilpotent element is a zero divisor: if $n$ is the least integer with $x^n = 0$, then $x \cdot x^{n-1} = 0$ and $x^{n-1} \neq 0$. An integral domain has no nilpotent element except $0$.

Explanations

L1 human

A nilpotent element is not zero, but some power of it is zero. Numbers do not behave like this. In the integers modulo $8$, the element $2$ does: $2 \cdot 2 \cdot 2 = 8$, and $8$ is $0$ modulo $8$.