Nilpotent

This article is about a type of element in a ring. For the type of group, seeNilpotent group.For the type of ideal, seeNilpotent ideal.For the type of semigroup, seeNilpotent semigroup.For the type of algebra, seeNilpotent algebra.

Inmathematics,an element$x$of aring$R$is callednilpotentif there exists some positiveinteger$n$,called theindex(or sometimes thedegree), such that$x^{n}=0$.

The term, along with its sisteridempotent,was introduced byBenjamin Peircein the context of his work on the classification of algebras.

Examples

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

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

is nilpotent because$A^{3}=0$.Seenilpotent matrixfor more.

• In thefactor ring$\mathbb {Z} /9\mathbb {Z}$,theequivalence classof 3 is nilpotent because 3²iscongruentto 0modulo9.
• Assume that two elements$a$and$b$in a ring$R$satisfy$ab=0$.Then the element$c=ba$is nilpotent as$${\begin{aligned}c^{2}&=(ba)^{2}\\&=b(ab)a\\&=0.\\\end{aligned}}$$An example with matrices (fora,b):$$A={\begin{pmatrix}0&1\\0&1\end{pmatrix}},\;\;B={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.$$Here$AB=0$and$BA=B$.

• By definition, any element of anilsemigroupis nilpotent.

Properties

No nilpotent element can be aunit(except in thetrivial ring,which has only a single element0 = 1). All nilpotent elements arezero divisors.

An$n\times n$matrix$A$with entries from afieldis nilpotent if and only if itscharacteristic polynomialis$t^{n}$.

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

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

Commutative rings

The nilpotent elements from acommutative ring$R$form anideal${\mathfrak {N}}$;this is a consequence of thebinomial theorem.This ideal is thenilradicalof the ring. Every nilpotent element$x$in a commutative ring is contained in everyprime ideal${\mathfrak {p}}$of that ring, since$x^{n}=0\in {\mathfrak {p}}$.So${\mathfrak {N}}$is contained in the intersection of all prime ideals.

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

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

Nilpotent elements in Lie algebra

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

Nilpotency in physics

Anyladder operatorin a finite dimensional space is nilpotent. They representcreation and annihilation operators,which transform from one state to another, for example the raising and loweringPauli matrices$\sigma _{\pm }=(\sigma _{x}\pm i\sigma _{y})/2$.

Anoperand$Q$that satisfies$Q^{2}=0$is nilpotent.Grassmann numberswhich allow apath integralrepresentation for Fermionic fields are nilpotents since their squares vanish. TheBRST chargeis an important example inphysics.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition.More generally, in view of the above definitions, an operator$Q$is nilpotent if there is$n\in \mathbb {N}$such that$Q^{n}=0$(thezero function). Thus, alinear mapis nilpotentiffit has a nilpotent matrix in some basis. Another example for this is theexterior derivative(again with$n=2$). Both are linked, also throughsupersymmetryandMorse theory,as shown byEdward Wittenin a celebrated article.

Theelectromagnetic fieldof a plane wave without sources is nilpotent when it is expressed in terms of thealgebra of physical space.More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is partsmooth infinitesimal analysis.

Algebraic nilpotents

The two-dimensionaldual numberscontain a nilpotent space. Other algebras and numbers that contain nilpotent spaces includesplit-quaternions(coquaternions),split-octonions, biquaternions$\mathbb {C} \otimes \mathbb {H}$,and complexoctonions$\mathbb {C} \otimes \mathbb {O}$.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.

See also

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