Non-associative ring

Definition
A non-associative ring, more properly called a possibly non-associative ring or a not necessarily associative ring, is defined as a set $$R$$ equipped with the following operations:


 * An infix binary operation $$+$$, called addition.
 * A prefix unary operation $$-$$, called the negative.
 * A constant element $$0$$, called zero.
 * A binary operation $$*$$, called the multiplication.

satisfying the following compatibility conditions:


 * $$R$$ forms an defining ingredient::abelian group with group operation $$+$$, inverse operation $$-$$, and identity element $$0$$.
 * $$R$$ satisfies the two distributivity laws:
 * $$a * (b + c) = (a * b) + (a * c) \ \forall \ a,b,c \in R$$
 * $$(a + b) * c = (a * c) + (b * c) \ \forall \ a,b,c \in R$$