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In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0.
Examples
This definition can be applied in particular to square matrices. The matrix
- <math>A = \begin{pmatrix}
0&1&0\\
0&0&1\\
0&0&0\end{pmatrix}
<math>
is nilpotent because A3 = 0. It can be shown that an n×n matrix A is nilpotent if and only if its characteristic polynomial is given by
- det(λI − A) = λn.
In the factor ring Z/9Z, the class of 3 is nilpotent because 32 is congruent to 0 modulo 9.
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.
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.
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.
If x is nilpotent, then 1-x is a unit, because xn = 0 entails
- (1-x) (1 + x + x2 + ... + xn-1) = 1 - xn = 1.
Nilpotency in physics
An operator <math>Q<math> that satisfies <math>Q^2=0<math> is nilpotent. The BRST charge is an important example in physics.
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