nilpotent element

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 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.

Revised on April 29, 2014 23:27:30
by Toby Bartels
(98.16.169.231)