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Concept
Nilpotent
Summary
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.
This definition can be applied in particular to square matrices. The matrix
is nilpotent because . See nilpotent matrix for more.
In the factor ring , the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9.
Assume that two elements and in a ring satisfy . Then the element is nilpotent as An example with matrices (for a, b): Here and .
By definition, any element of a nilsemigroup is nilpotent.
No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). 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.
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, 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 . 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.
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 ).
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