In mathematics, an element of a ring is called nilpotent if there exists some positive integer such that . The smallest such is called the index of nilpotency or the degree of nilpotency of .¹²

This article is about a type of element in a ring. For the type of group, see Nilpotent group. For the type of ideal, see Nilpotent ideal. For the type of semigroup, see Nilpotent semigroup. For the type of algebra, see Nilpotent algebra.

The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.³

Examples

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.

Properties

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.

Commutative rings

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. If , i.e., has no non-zero nilpotent elements, is called a reduced 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. Conversely, 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 .⁴ 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.⁵

A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of a 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 the commutator subalgebra and is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Nilpotency in physics

Any ladder operator in a finite dimensional space is nilpotent. They represent creation and annihilation operators, which transform from one state to another, for example the raising and lowering Pauli matrices .

An operand 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.⁶⁷ 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,⁸ as shown by in a celebrated article.

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.

References

1Bland, Paul E. (2011). Rings and Their Modules. De Gruyter. p. 13. ISBN 978-3-11-025022-0.
2Pal, Madhumangal (2013). Advanced Algebra. Delhi: PHI Learning Private Limited. p. 533. ISBN 978-81-203-4737-3.
3Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
4Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6.
5Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). "Chapter 1: Rings and Ideals". Introduction to Commutative Algebra. Westview Press. p. 5. ISBN 978-0-201-40751-8.
6Peirce, B. Linear Associative Algebra. 1870.
7Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0

Study aids for Nilpotent

Terminology

index of nilpotency: The smallest such is called the index of nilpotency or the degree of nilpotency of .
degree of nilpotency: The smallest such is called the index of nilpotency or the degree of nilpotency of .

Terms the article defines in bold, quoted in context.

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