nLab nilpotent element

Idea

An element $x$ in a ring (or potentially even a nonassociative rig) $A$ is nilpotent if there exist a natural number $n$ such that $x^n = 0$.

An ring/rig/algebra is nilpotent if there exists a uniform number $n$ such that any product of $n$ elements is $0$. An algebra over a field is locally nilpotent if all of its finitely-generated subalgebras are nilpotent. A Lie algebra element is ad-nilpotent if the multiplication with any of its elements is a nilpotent linear operator. A Lie algebra $A$ is nilpotent iff its lower central series $A, [A,A], [A,[A,A]], \ldots, [A,[A,[A,\ldots,[A,A]\cdots]]], \ldots$ terminates with $0$ after finitely many steps. By Engel’s theorem (English Wikipedia) a finite-dimensional Lie algebra is nilpotent iff it is locally nilpotent. Thus sometimes locally nilpotent Lie algebras are called Engel’s Lie algebras.

The class of locally nilpotent associative algebras is closed under extensions (defined in the category of associative algebras). Consequently associative algebras have a largest nilpotent ideal (namely the sum of all locally nilpotent ideals), which is called Levitskii radical.

The structure rings of classical algebraic varieties are finitely generated noetherian commutative associative unital rings without nilpotent elements. One of the principal advantages of Grothendieck’s theory of schemes is to allow for nilpotent elements in local rings. A scheme is reduced if there are no nilpotent elements in stalks of the structure sheaf.

Related pages

• nilpotent group

• nilpotent topological space

• nilradical

For Lie algebras:

• Jacobson-Morozov theorem