|
A non-associative algebra is an algebra in which the assumption of multiplicative associativity is dropped. From this definition, a non-associative algebra does not mean that the associativity fails. Rather, it enlarges the class of associative algebras, so that any associative algebra is a non-associative algebra.
In much of the literature concerning non-associative algebras, where the meaning of a ``non-associative algebra'' is clear, the word ``non-associative'' is dropped for simplicity and clarity.
Lie algebras and Jordan algebras are two famous examples of non-associative algebras that are not associative.
If we substitute the word ``algebra'' with ``ring'' in the above paragraphs, then we arrive at the definition of a non-associative ring. Alternatively, a non-associative ring is just a non-associative algebra over the integers.
- 1
- Richard D. Schafer, An Introduction to Nonassociative Algebras, Dover Publications, (1995).
|
