A non-associative ring, more properly called a possibly non-associative ring or a not necessarily associative ring, is defined as a set equipped with the following operations:
- An infix binary operation , called addition.
- A prefix unary operation , called the negative.
- A constant element , called zero.
- A binary operation , called the multiplication.
satisfying the following compatibility conditions:
- forms an abelian group with group operation , inverse operation , and identity element .
- satisfies the two distributivity laws: