Nilpotent element

Let $R$ be a ring. An element $a\in R$ is nilpotent if there exists an integer $n\ge 1$ such that $a^n=0$.

Nilpotent elements generate nil ideals and collectively form the nilradical in the commutative case. They measure the failure of a ring to be reduced and play a central role in deformation and infinitesimal structure.

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 $0$.