In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xⁿ = 0. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

Examples

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

is nilpotent because A³ = 0. See nilpotent matrix for more.

• In the factor ring Z/9Z, the class of 3 is nilpotent because 3² is congruent to 0 modulo 9.

• Assume that two elements a,b in a (non-commutative) ring R satisfy ab=0. Then the element c=ba is nilpotent (if non-zero) as c²=(ba)²=b(ab)a=0. An example with matrices (for a,b):

Here .

• The ring of coquaternions contains a cone of nilpotents.

For the square matrix section, see square matrix. ... In mathematics, a nilpotent matrix is a square matrix that is nilpotent. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... The word modulo is the Latin ablative of modulus. ... In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ... In common usage and elementary geometry, a cone (Greek: κώνος) is a solid object obtained by rotating a right triangle around one of its two short sides, the cones axis. ...

Properties

No nilpotent element can be a unit (except in the trivial ring {0} which has only a single element 0 = 1). All non-zero nilpotent elements are zero divisors. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... A trivial ring is a ring defined on a singleton set, {x}. The ring operations (* and +) are trivial: x * x = x x + x = x Clearly this ring is commutative. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...

An n-by-n matrix A with entries from a field is nilpotent if and only if its characteristic polynomial is Tⁿ, which is the case if and only if Aⁿ = 0. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...

The nilpotent elements from a commutative ring form an ideal; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element in a commutative ring is contained in every prime ideal of that ring, and in fact the intersection of all these prime ideals is equal to the nilradical. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...

If x is nilpotent, then 1 − x is a unit, because xⁿ = 0 entails In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

(1 − x) (1 + x + x² + ... + xⁿ⁻¹) = 1 − xⁿ = 1.

Nilpotency in physics

An operator Q that satisfies Q² = 0 is nilpotent. The BRST charge is an important example in physics. In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ... In theoretical physics, the BRST formalism is a method of implementing first class constraints. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ...

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∈N such that Qⁿ=o (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 n=2). Both are linked, also through supersymmetry and Morse theory, as shown by Edward Witten in a celebrated article. Zero redirects here. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification friend or foe - a battlefield identification system iff - the mathematics concept if and only if International Film Festival TIFF - Toronto International Film Festival Montreal International Film Festival This is a disambiguation... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In particle physics, supersymmetry is a hypothetical symmetry that relates bosons and fermions. ... In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ... Edward Witten at Harvard University Edward Witten (born August 26, 1951) is a professor at the Institute for Advanced Study and one of the leading researchers in string theory (M-theory). ...

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space. In the physics of electromagnetism, an electromagnetic field is a field composed of two related vector fields: the electric field and the magnetic field. ... This article may be too technical for most readers to understand. ...

References

• E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661-692,1982.
• A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703-3714,2000 DOI:10.1088/0264-9381/17/18/309.