In

^{2} is congruent to 0 modulo 9.
* Assume that two elements $a$ and $b$ in a ring $R$ satisfy $ab=0$. Then the element $c=ba$ is nilpotent as $$\backslash beginc^2\&=(ba)^2\backslash \backslash \; \&=b(ab)a\backslash \backslash \; \&=0.\backslash \backslash \; \backslash end$$ An example with matrices (for ''a'', ''b''):$$A\; =\; \backslash begin\; 0\; \&\; 1\backslash \backslash \; 0\; \&\; 1\; \backslash end,\; \backslash ;\backslash ;\; B\; =\backslash begin\; 0\; \&\; 1\backslash \backslash \; 0\; \&\; 0\; \backslash end.$$ Here $AB=0$ and $BA=B$.
*By definition, any element of a nilsemigroup is nilpotent.

mathematics
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, an element $x$ of a ring $R$ is called nilpotent if there exists some positive integer
An integer is the number zero (), a positive natural number
In mathematics, the natural numbers are those number
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$n$, called the index (or sometimes the degree), such that $x^n=0$.
The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.
Examples

*This definition can be applied in particular to square matrices. The matrix :: $A\; =\; \backslash begin\; 0\; \&\; 1\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 1\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \backslash end$ :is nilpotent because $A^3=0$. See nilpotent matrix for more. * In the factor ring $\backslash Z/9\backslash Z$, theequivalence class
In mathematics
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of 3 is nilpotent because 3Properties

No nilpotent element can be a unit (except in the trivial ring, which has only a single element ). All nilpotent elements are zero divisors. An $n\backslash times\; n$ matrix $A$ with entries from a field is nilpotent if and only if its characteristic polynomial is $t^n$. If $x$ is nilpotent, then $1-x$ is a unit, because $x^n=0$ entails $$(1\; -\; x)\; (1\; +\; x\; +\; x^2\; +\; \backslash cdots\; +\; x^)\; =\; 1\; -\; x^n\; =\; 1.$$ More generally, the sum of a unit element and a nilpotent element is a unit when they commute.Commutative rings

The nilpotent elements from a commutative ring $R$ form an ideal $\backslash mathfrak$; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element $x$ in a commutative ring is contained in every prime ideal $\backslash mathfrak$ of that ring, since $x^n\; =\; 0\backslash in\; \backslash mathfrak$. So $\backslash mathfrak$ is contained in the intersection of all prime ideals. If $x$ is not nilpotent, we are able to localize with respect to the powers of $x$: $S=\backslash $ to get a non-zero ring $S^R$. The prime ideals of the localized ring correspond exactly to those prime ideals $\backslash mathfrak$ of $R$ with $\backslash mathfrak\backslash cap\; S=\backslash empty$. 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 $\backslash mathfrak$ is exactly the intersection of all prime ideals. A characteristic similar to that ofJacobson radical In mathematics
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and 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 $\backslash mathfrak$ be a Lie algebra. Then an element of $\backslash mathfrak$ is called nilpotent if it is in $;\; href="/html/ALL/s/mathfrak,\_\backslash mathfrak.html"\; ;"title="mathfrak,\; \backslash mathfrak">mathfrak,\; \backslash mathfrak$Nilpotency in physics

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 $\backslash sigma\_\backslash pm=(\backslash sigma\_x\backslash pm\; i\; \backslash sigma\_y)/2$. An operand $Q$ that satisfies $Q^2=0$ 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 inphysics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

.
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 $n\backslash in\backslash N$ such that $Q^n=0$ (the zero function). Thus, a linear map
In mathematics
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is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with $n=2$). Both are linked, also through supersymmetry and Morse theory, as shown by Edward Witten in a celebrated article.
The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.Rowlands, P. ''Zero to Infinity: The Foundations of Physics'', London, World Scientific 2007, 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.
Algebraic nilpotents

The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions $\backslash mathbb\; C\backslash otimes\backslash mathbb\; H$, and complex octonions $\backslash mathbb\; C\backslash otimes\backslash mathbb\; O$. 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 In ring theory, a branch of mathematics
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* Nil ideal
References

{{Reflist Ring theory 0 (number) Algebraic properties of elements