In
mathematics, an element
of a
ring is called nilpotent if there exists some positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, called the index (or sometimes the degree), such that
.
The term was introduced by
Benjamin Peirce
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philos ...
in the context of his work on the classification of algebras.
Examples
*This definition can be applied in particular to
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
. The matrix
::
:is nilpotent because
. See
nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that
:N^k = 0\,
for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N.
More generally, a nilpotent transformation is a linear transf ...
for more.
* In the
factor ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
, the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of 3 is nilpotent because 3
2 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
In mathematics, and more precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent.
Definitions
Formally, a semigroup ''S'' is a nilsemigroup if:
*''S'' contains ''0'' and
*for each eleme ...
is nilpotent.
Properties
No nilpotent element can be a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
(except in the
trivial ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
, which has only a single element ). All nilpotent elements are
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s.
An
matrix
with entries from a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is nilpotent if and only if its
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The char ...
is
.
If
is nilpotent, then
is a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
, because
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
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
form an
ideal ; this is a consequence of the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. This ideal is the
nilradical of the ring. Every nilpotent element
in a commutative ring is contained in every
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
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 In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definition ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. Then an element of
is called nilpotent if it is in