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.
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.[1]
^Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
an element x {\displaystyle x} of a ring R {\displaystyle R} is called nilpotent if there exists some positive integer n {\displaystyle n} , called the...
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} ...
In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has...
more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k = 0. By I k, it...
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive...
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple...
in a commutative ring A is locally nilpotent at a prime ideal p if Ip, the localization of I at p, is a nilpotent ideal in Ap. In non-commutative algebra...
In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower...
In topology, a branch of mathematics, a nilpotent space, first defined by Emmanuel Dror (1969), is a based topological space X such that the fundamental...
trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in...
a Hilbert space is said to be nilpotent if Tn = 0 for some n. It is said to be quasinilpotent or topologically nilpotent if its spectrum σ(T) = {0}. In...
that H has property P. Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the...
In mathematics, the nilpotent cone N {\displaystyle {\mathcal {N}}} of a finite-dimensional semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} is...
More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent. In particular, the...
{\displaystyle R/I} -bimodule. More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient R → R r e d {\displaystyle...
mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example...
whose elements are nilpotent elements. 3. A nilpotent ideal is an ideal whose power Ik is {0} for some positive integer k. Every nilpotent ideal is nil, but...
precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent. Formally, a semigroup S is a nilsemigroup...
groups are all abelian. A nilpotent series is a subnormal series such that successive quotients are nilpotent. A nilpotent series exists if and only if...
group theory, the Fitting length (or nilpotent length) measures how far a solvable group is from being nilpotent. The concept is named after Hans Fitting...
theory, and solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable...
the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: N R = { f ∈ R ∣ f m = 0 for some m ∈ Z > 0 } . {\displaystyle...
its Lie algebra. The theory was introduced by Kirillov (1961, 1962) for nilpotent groups and later extended by Bertram Kostant, Louis Auslander, Lajos Pukánszky...
products involving nilpotent elements and sums of nilpotent elements are both nilpotent. This is because if a and b are nilpotent elements of R with an...
\partial } of a commutative ring A{\displaystyle A} is called a locally nilpotent derivation (LND) if every element of A{\displaystyle A} is annihilated...
with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension 3 {\displaystyle 3} , playing a key role in quantum...
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