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
An associative ring, sometimes termed a ring, is 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:
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 . 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.