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''.Nil algebra – Encyclopedia of Mathematics
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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

*{{Lang Algebra

External links

Nilpotent algebra – Encyclopedia of Mathematics


Ring theory
Properties of binary operations