Nilpotent

In mathematics, an element "x" of a ring "R" is called nilpotent if there exists some positive integer "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.
Properties.
No nilpotent element can be a unit (except in the trivial ring {0}, which has only a single element ). All non-zero nilpotent elements are zero divisors.
An "n"-by-"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
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 formula_4 form an ideal formula_5; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element formula_6 in a commutative ring is contained in every prime ideal formula_7 of that ring, since formula_8. So formula_5 is contained in the intersection of all prime ideals.
If formula_6 is not nilpotent, we are able to localize with respect to the powers of formula_6: formula_12 to get a non-zero ring formula_13. The prime ideals of the localized ring correspond exactly to those prime ideals formula_7 of formula_4 with formula_16. As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent formula_6 is not contained in some prime ideal. Thus formula_5 is exactly the intersection of all prime ideals.
A characteristic similar to that of Jacobson radical 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 formula_19 be a Lie algebra. Then an element of formula_19 is called nilpotent if it is in formula_21 and formula_22 is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.
Nilpotency in physics.
An operand "Q" that satisfies 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. More generally, in view of the above definitions, an operator "Q" is nilpotent if there is such that "Q""n" = 0 (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 ). 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 used to derive theorems 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 formula_23, and complex octonions formula_24.