De Branges's theorem: Difference between revisions

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{{about|a type of element in a ring|the type of group|Nilpotent group|the type of ideal|Nilpotent ideal}}


In [[mathematics]], an element ''x'' of a [[ring (mathematics)|ring]] ''R'' is called '''nilpotent''' if there exists some positive [[integer]] ''n'' such that ''x''<sup>''n''</sup>&nbsp;=&nbsp;0.


The term was introduced by [[Benjamin Peirce]]<ref>Polcino & Sehgal (2002), p. 127.</ref> in the context of elements of an algebra that vanish when raised to a power.
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== Examples ==
 
*This definition can be applied in particular to [[square matrix|square matrices]]. The matrix
:: <math>A = \begin{pmatrix}
0&1&0\\
0&0&1\\
0&0&0\end{pmatrix}
</math>
:is nilpotent because ''A''<sup>3</sup> =&nbsp;0. See [[nilpotent matrix]] for more.
 
*In the [[factor ring]] '''Z'''/9'''Z''', the equivalence class of 3 is nilpotent because 3<sup>2</sup> is congruent to 0 [[Modular arithmetic|modulo]] 9.
 
* Assume that two elements ''a'',&nbsp;''b'' in a (non-commutative) ring ''R'' satisfy ''ab''&nbsp;=&nbsp;0. Then the element ''c''&nbsp;=&nbsp;''ba'' is nilpotent (if non-zero) as ''c''<sup>2</sup>&nbsp;=&nbsp;(''ba'')<sup>2</sup>&nbsp;=&nbsp;''b''(''ab'')''a''&nbsp;=&nbsp;0. An example with matrices (for ''a'',&nbsp;''b''):
 
:: <math>A = \begin{pmatrix}
0&1\\
0&1
\end{pmatrix}, \;\;
B =\begin{pmatrix}
0&1\\
0&0
\end{pmatrix}.
</math>
: Here ''AB''&nbsp;=&nbsp;0, ''BA''&nbsp;=&nbsp;''B''.
 
*The ring of [[coquaternion]]s contains a [[cone (linear algebra)|cone]] of nilpotents.
 
== Properties ==
No nilpotent element can be a [[unit (ring theory)|unit]] (except in the [[trivial ring]] {0} which has only a single element 0 = 1). All non-zero nilpotent elements are [[zero divisor]]s.
 
An ''n''-by-''n'' matrix ''A'' with entries from a [[field (mathematics)|field]] is nilpotent if and only if its [[characteristic polynomial]] is ''t''<sup>''n''</sup>.
 
If ''x'' is nilpotent, then 1&nbsp;&minus;&nbsp;''x'' is a [[Unit (ring theory)|unit]], because ''x''<sup>''n''</sup> =&nbsp;0 entails
:<math>(1 - x) (1 + x + x^2 + \cdots + x^{n-1}) = 1 - x^n = 1.\ </math>
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]] <math>R</math> form an [[ring ideal|ideal]] <math>\mathfrak{N}</math>; this is a consequence of the [[binomial theorem]]. This ideal is the [[Nilradical of a ring|nilradical]] of the ring. Every nilpotent element <math>x</math> in a commutative ring is contained in every [[prime ideal]] <math>\mathfrak{p}</math> of that ring, since <math>x^n=0\in \mathfrak{p}</math>. So <math>\mathfrak{N}</math> is contained in the intersection of all prime ideals.
 
If <math>x</math> is not nilpotent, we are able to [[localization of a ring#Construction and properties for commutative rings|localize]] with respect to the powers of <math>x</math>: <math>S=\{1,x,x^2,...\}</math> to get a non-zero ring <math>S^{-1}R</math>. The prime ideals of the localized ring correspond exactly to those primes <math>\mathfrak{p}</math> with <math>\mathfrak{p}\cap S=\empty</math>.<ref>{{cite book |last=Matsumura |first=Hideyuki |title=Commutative Algebra |publisher=W. A. Benjamin |year=1970 |pages=6 |chapter=Chapter 1: Elementary Results |isbn=978-0-805-37025-6}}</ref> As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent <math>x</math> is not contained in some prime ideal. Thus <math>\mathfrak{N}</math> is exactly the intersection of all prime ideals.<ref>{{cite book |last1=Atiyah |first1=M. F. | last2=MacDonald |first2=I. G. | title=Introduction to Commutative Algebra |publisher=Westview Press |date=February 21, 1994 |pages=5 |chapter=Chapter 1: Rings and Ideals |isbn=978-0-201-40751-8}}</ref>
 
== Nilpotent elements in Lie algebra ==
Let <math>\mathfrak{g}</math> be a Lie algebra. Then an element of <math>\mathfrak{g}</math> is called nilpotent if it is in <math>[\mathfrak{g}, \mathfrak{g}]</math> and <math>\operatorname{ad} x</math> is a nilpotent transformation. See also: [[Jordan decomposition in a Lie algebra]].
 
==Nilpotency in physics==
 
An [[operand]] ''Q'' that satisfies ''Q''<sup>2</sup>&nbsp;=&nbsp;0 is nilpotent. [[Grassmann number]]s which allow a [[Path integral formulation|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.<ref>Peirce, B. ''Linear Associative Algebra''. 1870.</ref><ref>Milies, César Polcino; Sehgal, Sudarshan K. ''An introduction to group rings''. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0</ref> More generally, in view of the above definitions, an operator ''Q'' is nilpotent if there is ''n''∈'''N''' such that ''Q''<sup>''n''</sup>&nbsp;=&nbsp;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 ''n''&nbsp;=&nbsp;2). Both are linked, also through [[supersymmetry]] and [[Morse theory]],<ref>A. Rogers, ''The topological particle and Morse theory'',  Class. Quantum Grav. 17:3703–3714,2000 {{doi|10.1088/0264-9381/17/18/309}}.</ref> as shown by [[Edward Witten]] in a celebrated article.<ref>E Witten, ''Supersymmetry and Morse theory''. J.Diff.Geom.17:661–692,1982.</ref>
 
The [[electromagnetic field]] of a plane wave without sources is nilpotent when it is expressed in terms of the [[algebra of physical space]].<ref>Rowlands, P. ''Zero to Infinity: The Foundations of Physics'', London, World Scientific 2007, ISBN 978-981-270-914-1</ref>
 
==Algebraic nilpotents==
The two-dimensional [[dual number]]s contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include [[split-quaternion]]s (coquaternions), [[split-octonion]]s,
[[biquaternion]]s <math>\mathbb C\otimes\mathbb H</math>, and complex [[octonions]] <math>\mathbb C\otimes\mathbb O</math>.
 
==See also==
*[[Idempotent element]]
*[[Unipotent]]
*[[Reduced ring]]
*[[Nil ideal]]
 
== References ==
{{Reflist}}
 
[[Category:Ring theory]]
[[Category:Zero]]

Latest revision as of 14:53, 24 November 2014


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