A ring is an algebraic structure (some definitions say a set) in which two binary operators (addition and multiplication) in various combinations must satisfy either the additive associative, commutative, identity, and inverse properties, the multiplicative associative property, or the left and right distributivity properties. For example, the elements of one operation, such as addition, must form a group that is commutative, also known as an abelian group. The multiplicative operation must produce unique answers that have the associative property. These two operations are further connected by requiring the multiplication to have a distributive property with respect to the addition. This can be written as follows, with a, b, and c elements of the ring:

Rings are usually named after one or more of their investigators. But such a practice usually makes understanding the properties of the various associated rings difficult for anyone other than the mathematician working on the ring.