Nilpotent on arvau

Wikipedia / Nilpotent

In mathematics of a ring is called nilpotent if there exists some positive integer is called the index of nilpotency or the degree of nilpotency.

The term, along with its sister idempotent, 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
. See nilpotent matrix for more.
In the factor ring, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9.
An example with matrices (for a,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.

with entries from a field is nilpotent if and only if its characteristic polynomial.

is a unit

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 form an ideal; this is a consequence of the binomial theorem. This ideal is the nilradical is called a reduced ring.

in a commutative ring is contained in every prime ideal is not nilpotent, we are able to localize. As every non-zero commutative ring has a maximal ideal is exactly the intersection of all prime ideals.

A characteristic similar to that of Jacobson radical and annihilation of simple modules are precisely those that annihilate all integral domains). This follows from the fact that nilradical is the intersection of all prime ideals.

Nilpotent elements in Lie algebra

be a Lie algebra 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.

An operand 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.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. (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). 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. More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is part of 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, and complex octonions. 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.