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, a concept related to quantum groups and Hopf algebras.

Formal definition
An associative algebra $$A$$ over a commutative ring $$R$$ is defined to be a nilpotent algebra if and only if there exists some positive integer $$n$$ such that $$0=y_1\ y_2\ \cdots\ y_n$$ for all $$y_1, \  y_2, \ \ldots,\ y_n$$ in the algebra $$A$$. The smallest such $$n$$ is called the index of the algebra $$A$$. In the case of a non-associative algebra, the definition is that every different multiplicative association of the $$n$$ elements is zero.

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

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.