In mathematics, an element ${\displaystyle x}$ of a ring ${\displaystyle R}$ is called nilpotent if there exists some positive integer ${\displaystyle n}$ such that ${\displaystyle x^{n}=0}$. The smallest such ${\displaystyle n}$ is called the index of nilpotency or the degree of nilpotency of ${\displaystyle x}$.

The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.

No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). All nilpotent elements are zero divisors.

An ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ with entries from a field is nilpotent if and only if its characteristic polynomial is ${\displaystyle t^{n}}$.

If ${\displaystyle x}$ is nilpotent, then ${\displaystyle 1-x}$ is a unit, because ${\displaystyle x^{n}=0}$ entails $${\displaystyle (1-x)(1+x+x^{2}+\cdots +x^{n-1})=1-x^{n}=1.}$$

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

The nilpotent elements from a commutative ring ${\displaystyle R}$ form an ideal ${\displaystyle {\mathfrak {N}}}$; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. If ${\displaystyle {\mathfrak {N}}=\{0\}}$, i.e., ${\displaystyle R}$ has no non-zero nilpotent elements, ${\displaystyle R}$ is called a reduced ring.

Every nilpotent element ${\displaystyle x}$ in a commutative ring is contained in every prime ideal ${\displaystyle {\mathfrak {p}}}$ of that ring, since ${\displaystyle x^{n}=0\in {\mathfrak {p}}}$. So ${\displaystyle {\mathfrak {N}}}$ is contained in the intersection of all prime ideals. Conversely, if ${\displaystyle x}$ is not nilpotent, we are able to localize with respect to the powers of ${\displaystyle x}$: ${\displaystyle S=\{1,x,x^{2},...\}}$ to get a non-zero ring ${\displaystyle S^{-1}R}$. The prime ideals of the localized ring correspond exactly to those prime ideals ${\displaystyle {\mathfrak {p}}}$ of ${\displaystyle R}$ with ${\displaystyle {\mathfrak {p}}\cap S=\emptyset }$. As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent ${\displaystyle x}$ is not contained in some prime ideal. Thus ${\displaystyle {\mathfrak {N}}}$ is exactly the intersection of all prime ideals.