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.
is defined to be a nilpotent algebra if and only if there exists some positive integer
is called the index of the algebra
[2] In the case of a non-associative algebra, the definition is that every different multiplicative association of the
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.