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.
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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.