Nilpotent element - Knowlpedia

Let $R$ be a ring. An element $a\in R$ is nilpotent if $a^n=0$ for some integer $n\ge 1$ .

Remarks

In a commutative ring, the nilpotent elements form the nilradical, and the ring is reduced exactly when $0$ is its only nilpotent element. For a noncommutative ring, the nilpotent elements need not form an ideal, and the ideal generated by a nilpotent element need not be a nil ideal.

Examples

In $k[x]/(x^n)$ , the class of $x$ is nilpotent.
Any strictly upper triangular matrix is nilpotent.
In an integral domain, the only nilpotent element is $0$ .