Nilpotent

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

The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.^[1]

• This definition can be applied in particular to square matrices. The matrix

${\displaystyle A={\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}}}$

is nilpotent because ${\displaystyle A^{3}=0}$. See nilpotent matrix for more.

• In the factor ring ${\displaystyle \mathbb {Z} /9\mathbb {Z} }$, the equivalence class of 3 is nilpotent because 3² is congruent to 0 modulo 9.
• Assume that two elements ${\displaystyle a}$ and ${\displaystyle b}$ in a ring ${\displaystyle R}$ satisfy ${\displaystyle ab=0}$. Then the element ${\displaystyle c=ba}$ is nilpotent as $${\displaystyle {\begin{aligned}c^{2}&=(ba)^{2}\\&=b(ab)a\\&=0.\\\end{aligned}}}$$ An example with matrices (for a, b):$${\displaystyle A={\begin{pmatrix}0&1\\0&1\end{pmatrix}},\;\;B={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.}$$ Here ${\displaystyle AB=0}$ and ${\displaystyle BA=B}$.

• By definition, any element of a nilsemigroup is nilpotent.

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. 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.

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 ${\displaystyle {\mathfrak {p}}\cap S=\emptyset }$.^[2] 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.^[3]

A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of a ring ${\displaystyle R}$ are precisely those that annihilate all integral domains internal to the ring ${\displaystyle R}$ (that is, of the form ${\displaystyle R/I}$ for prime ideals ${\displaystyle I}$). This follows from the fact that nilradical is the intersection of all prime ideals.

Let ${\displaystyle {\mathfrak {g}}}$ be a Lie algebra. Then an element ${\displaystyle x\in {\mathfrak {g}}}$ is called nilpotent if it is in ${\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]}$ and ${\displaystyle \operatorname {ad} x}$ is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Any ladder operator in a finite dimensional space is nilpotent. They represent creation and annihilation operators, which transform from one state to another, for example the raising and lowering Pauli matrices ${\displaystyle \sigma _{\pm }=(\sigma _{x}\pm i\sigma _{y})/2}$.

An operand ${\displaystyle Q}$ that satisfies ${\displaystyle Q^{2}=0}$ is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. The BRST charge is an important example in physics.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition.^[4][5] More generally, in view of the above definitions, an operator ${\displaystyle Q}$ is nilpotent if there is ${\displaystyle n\in \mathbb {N} }$ such that ${\displaystyle Q^{n}=0}$ (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with ${\displaystyle n=2}$). Both are linked, also through supersymmetry and Morse theory,^[6] as shown by Edward Witten in a celebrated article.^[7]

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.^[8] More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is part smooth infinitesimal analysis.

The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions ${\displaystyle \mathbb {C} \otimes \mathbb {H} }$, and complex octonions ${\displaystyle \mathbb {C} \otimes \mathbb {O} }$. If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an indefinitely small proportion of the first order term.

• Idempotent element (ring theory)
• Unipotent
• Reduced ring
• Nil ideal
• Nilpotent matrix