Associative ring

This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.
View other non-associative ring properties

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 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.