nLab nonassociative algebra




Let kk be a commutative unital ring, usually a field (but conceivably even a commutative rig).

A nonassociative kk-algebra is a kk-module VV equipped with a bilinear product VVVV\otimes V\to V.

This product is typically neither associative nor unital, although it can be (an example of the red herring principle).

Mathematicians working in the field of nonassociative algebras often say simply ‘algebra’ meaning a nonassociative algebra.


Some interesting subclasses are Lie algebra, Jordan algebra, Leibniz algebra, alternative algebra, associative unital algebra, composition algebra

The octonions are a (slightly) non-associative real normed division algebra.


  • Richard D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995. (pdf)

Last revised on April 21, 2017 at 07:42:45. See the history of this page for a list of all contributions to it.