Nilpotent Explained
In mathematics, an element
of a
ring
is called
nilpotent if there exists some positive
integer
, called the
index (or sometimes the
degree), such that
.
The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.[1]
Examples
- This definition can be applied in particular to square matrices. The matrix
A=\begin{pmatrix}
0&1&0\\
0&0&1\\
0&0&0
\end{pmatrix}
is nilpotent because
. See
nilpotent matrix for more.
, the
equivalence class of 3 is nilpotent because 3
2 is
congruent to 0
modulo 9.
and
in a ring
satisfy
. Then the element
is nilpotent as
An example with matrices (for
a,
b):
Here
and
.
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
matrix
with entries from a
field is nilpotent if and only if its
characteristic polynomial is
.
If
is nilpotent, then
is a
unit, because
entails
More generally, the sum of a unit element and a nilpotent element is a unit when they commute.
Commutative rings
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, since
. So
is contained in the intersection of all prime ideals.
If
is not nilpotent, we are able to localize with respect to the powers of
:
to get a non-zero ring
. The prime ideals of the localized ring correspond exactly to those prime ideals
of
with
.
[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent
is not contained in some prime ideal. Thus
is exactly the intersection of all prime ideals.
[3] A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of ring
are precisely those that annihilate all integral domains internal to the ring
(that is, of the form
for prime ideals
). This follows from the fact that nilradical is the intersection of all prime ideals.
Nilpotent elements in Lie algebra
Let
be a
Lie algebra. Then an element
is called nilpotent if it is in
and
is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.
Nilpotency in physics
\sigma\pm=(\sigmax\pmi\sigmay)/2
.
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.[4] [5] More generally, in view of the above definitions, an operator
is nilpotent if there is
such that
(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,
[6] as shown by
Edward Witten in a celebrated article.
[7] The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.[8] 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
, 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.
See also
Notes and References
- Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
- Book: Matsumura, Hideyuki . Commutative Algebra . W. A. Benjamin . 1970 . 6 . Chapter 1: Elementary Results . 978-0-805-37025-6.
- Book: Atiyah . M. F. . MacDonald . I. G. . Introduction to Commutative Algebra . Westview Press . February 21, 1994 . 5 . Chapter 1: Rings and Ideals . 978-0-201-40751-8.
- Peirce, B. Linear Associative Algebra. 1870.
- Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002.
- A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714, 2000 .
- E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
- Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007,