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,^[1] a concept related to quantum groups and Hopf algebras.

Formal definition

An associative algebra ${\displaystyle A}$ over a commutative ring ${\displaystyle R}$ is defined to be a nilpotent algebra if and only if there exists some positive integer ${\displaystyle n}$ such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 0=y_1\ y_2\ \cdots\ y_n} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle y_1, \ y_2, \ \ldots,\ y_n} in the algebra ${\displaystyle A}$. The smallest such ${\displaystyle n}$ is called the index of the algebra ${\displaystyle A}$.^[2] In the case of a non-associative algebra, the definition is that every different multiplicative association of the ${\displaystyle n}$ 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