Nilpotent

Last updated January 27, 2022

Let $A$ be a ring.

Definition. An element $a\in A$ in nilpotent if there exists $N\in\mathbb{N}$ such that $a^N=0$.

Definition. The elements of a ring form an ideal $Nil(A)$ called the nilradical.

See Radical for more information.

Properties

1. A nilpotent element is a zero divisor, but the converse is not true in general.

• If $a$ and $b$ are nilpotent then so is $ab$ and $a+b$.
• The sum of a nilpotent element and a unit is a unit.