nilpotent element

An element $x$ in a nonassociative algebra $A$ is **nilpotent** if there exist a natural number $n$ such that $x^n = 0$.

An 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 its finitely-generated subalgebras are nilpotent. A Lie algebra 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 $A$, $[A,A]$, $[A,[A,A]]$, … ,$[A,[A,[A,\ldots[A,A]\ldots]]]$, … 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.

Revised on January 16, 2010 18:05:13
by Mike Shulman
(75.3.151.132)