# non-associative algebra

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 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.

## References

Title non-associative algebra NonassociativeAlgebra 2013-03-22 15:06:44 2013-03-22 15:06:44 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 17A01 Semifield Algebras non-associative ring