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.
Formal definition
An associative algebra over a commutative ring is defined to be a nilpotent algebra if and only if there exists some positive integer such that for all in the algebra . The smallest such 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 elements is zero.
Nil algebra
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.
See also
- Algebraic structure (a much more general term)
- nil-Coxeter algebra
- Lie algebra
- Example of a non-associative algebra
References
External links
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