Associative ring

Definition
An associative ring, sometimes termed a ring, is 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:
 * $$a * (b + c) = (a * b) + (a * c) \ \forall \ a,b,c \in R$$
 * $$(a + b) * c = (a * c) + (b * c) \ \forall \ a,b,c \in R$$
 * Associativity: $$a * (b * c) = (a * b) * c \ \forall \ a,b,c \in R$$

However, in many contexts, it is useful to study the situation where $$*$$ is possibly non-associative, i.e., we want to remove the last condition from the definition. We use the term non-associative ring for a ring that is not necessarily associative. Note that associative rings are non-associative rings by this definition.

A unital ring or unitary ring is a ring with an identity for multiplication, denoted $$1$$. A commutative ring is a ring where the multiplication is commutative. A commutative unital ring is a ring where the multiplication is both commutative and has a unit.