Nilpotent algebra

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 ${\displaystyle 0=y_{1}\ y_{2}\ \cdots \ y_{n}}$ for all ${\displaystyle y_{1},\ y_{2},\ \ldots ,\ y_{n}}$ 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.

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.

• Algebraic structure (a much more general term)
• nil-Coxeter algebra
• Lie algebra
• Example of a non-associative algebra

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

• Nilpotent algebra – Encyclopedia of Mathematics