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
- 1 Richard D. Schafer, An Introduction to Nonassociative Algebras, Dover Publications, (1995).