Nilpotent
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an element x of a
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
R is called nilpotent if there exists some positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
, was introduced by
Benjamin Peirce Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
in the context of his work on the classification of algebras.


Examples

*This definition can be applied in particular to
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See
nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear trans ...
for more. * In the
factor ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space (linear algebra), quo ...
\Z/9\Z, the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of 3 is nilpotent because 32 is
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
to 0
modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
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 Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
(except in the
trivial ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which fo ...
, which has only a single element ). All nilpotent elements are
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s. An n\times n matrix A with entries from a field is nilpotent if and only if its
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
is t^n. If x is nilpotent, then 1-x is a
unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
, 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 In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
R form an ideal \mathfrak; this is a consequence of the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
. This ideal is the nilradical of the ring. If \mathfrak=\, i.e., R has no non-zero nilpotent elements, R is called a
reduced ring In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x''  ...
. Every nilpotent element x in a commutative ring is contained in every
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
\mathfrak of that ring, since x^n = 0\in \mathfrak. So \mathfrak is contained in the intersection of all prime ideals. Conversely, 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 In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
, 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, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
and annihilation of
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
s is available for nilradical: nilpotent elements of a ring R are precisely those that annihilate all
integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
s 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
. Then an element x\in\mathfrak is called nilpotent if it is in the commutator subalgebra 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 Creation operators and annihilation operators are Operator (mathematics), mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilatio ...
, which transform from one state to another, for example the raising and lowering
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
\sigma_\pm=(\sigma_x\pm i \sigma_y)/2. An
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Unknown operands in equalities of expressions can be found by equation solving. Example The following arithmetic expres ...
Q that satisfies Q^2=0 is nilpotent.
Grassmann number In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known a ...
s 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 scientific study of matter, its Elementary particle, 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 whi ...
. 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 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
). Thus, a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
is nilpotent
iff In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...
it has a nilpotent matrix in some basis. Another example for this is the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
(again with n=2). Both are linked, also through
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
and
Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
, as shown by
Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
in a celebrated article. The
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
of a plane wave without sources is nilpotent when it is expressed in terms of the
algebra of physical space Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
.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 number In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. D ...
s contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include
split-quaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in t ...
s (coquaternions),
split-octonion In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signature (quadratic form), signatures of their ...
s,
biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions cor ...
s \mathbb C\otimes\mathbb H, and complex
octonions In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
\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) Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
*
Unipotent In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipote ...
*
Nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if all of its elements is nilpotent, i.e for each a \in I exists natural number ''n'' for which a^n = 0. If all elements of a ring ...


References

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