# Associative ring

From Groupprops

## Definition

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:
- Associativity:

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.