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