Ring

An algebraic structure (R, +, ·) consisting of a set R with two binary operations: addition + making (R, +) an abelian group, and multiplication · that is associative and distributes over addition. A multiplicative identity may or may not be required; commutativity of multiplication is optional.

An algebraic structure (R, +, ·) with two binary operations where (R, +) is an abelian group, multiplication · is associative, and · distributes over +; multiplication may or may not have an identity or be commutative depending on the convention.

A set equipped with two binary operations, typically called addition and multiplication, where addition forms an abelian group, multiplication is associative, and multiplication distributes over addition; rings may or may not have a multiplicative identity and need not be commutative under multiplication.

An associative algebraic structure consisting of a set equipped with two binary operations, addition and multiplication, where addition forms an abelian group, multiplication is associative, and multiplication distributes over addition; a multiplicative identity may or may not be assumed depending on convention.