Nilpotent algebra

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 integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring.) Another possible source of confusion in terminology is the quantum nilpotent algebra,¹ a concept related to quantum groups and Hopf algebras.

An associative algebra ${\displaystyle A}$ over a commutative ring ${\displaystyle R}$ is defined to be a nilpotent algebra if and only if there exists some positive integer ${\displaystyle n}$ such that ${\displaystyle 0=y_{1}\ y_{2}\ \cdots \ y_{n}}$ for all ${\displaystyle y_{1},\ y_{2},\ \ldots ,\ y_{n}}$ in the algebra ${\displaystyle A}$. The smallest such ${\displaystyle n}$ is called the index of the algebra ${\displaystyle A}$.² In the case of a non-associative algebra, the definition is that every different multiplicative association of the ${\displaystyle n}$ elements is zero.

A power associative algebra in which every element of the algebra is nilpotent is called a nil algebra.³

Nilpotent algebras are trivially nil, whereas nil algebras may not be nilpotent, as each element being nilpotent does not force products of distinct elements to vanish.