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 formula_1 over a commutative ring formula_2 is defined to be a nilpotent algebra if and only if there exists some positive integer formula_3 such that formula_4 for all formula_5 in the algebra formula_1. The smallest such formula_3 is called the index of the algebra formula_1. In the case of a non-associative algebra, the definition is that every different multiplicative association of the formula_3 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.
References.
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