Neha Patil (Editor)

Nilpotent algebra

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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.

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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   ⋯   y n for all y 1 ,   y 2 ,   … ,   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

An algebra in which every element of the algebra is nilpotent is called a nil algebra.

References

Nilpotent algebra Wikipedia
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