Nilpotent
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 A number is a mathematical object used to count, measure, and label. The original examples are the natural number ...
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 = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the
equivalence class In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
of 3 is nilpotent because 32 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 \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ 0 & 0 \end. Here AB=0 and BA=B. *By definition, any element of a nilsemigroup is nilpotent.


Properties

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\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 + \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 \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 \mathfrak of that ring, since x^n = 0\in \mathfrak. So \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=\ to get a non-zero ring S^R. The prime ideals of the localized ring correspond exactly to those prime ideals \mathfrak of R with \mathfrak\cap S=\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 \mathfrak is exactly the intersection of all prime ideals. A characteristic similar to that of
Jacobson radical In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
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 \mathfrak be a Lie algebra. Then an element of \mathfrak is called nilpotent if it is in mathfrak, \mathfrak/math> and \operatorname x is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.


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 \sigma_\pm=(\sigma_x\pm i \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 in
physics 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\in\N such that Q^n=0 (the zero function). Thus, a
linear map In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 \mathbb C\otimes\mathbb H, and complex octonions \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 In ring theory, a branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topi ...
* Nil ideal


References

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