Definition:Ring (Abstract Algebra)

A ring $$\left({R, *, \circ}\right)$$ is a semiring in which $$\left({R, *}\right)$$ forms a group.

That is, in addition to $$\left({R, *}\right)$$ being closed and associative under $$*$$, it also has an identity, and each element has an inverse.

Ring Axioms
A ring is an algebraic structure $$\left({R, *, \circ}\right)$$, on which is defined two binary operations $$\circ$$ and $$*$$, which satisfies the following conditions:

These four stipulations are called the ring axioms.

Note that a ring is still a semiring, so all properties of a semiring also apply to a ring.

Addition
The distributand $$*$$ of a ring $$\left({R, *, \circ}\right)$$ is referred to as addition.

The conventional for this operation is $$+$$, and a general ring is frequently denoted $$\left({R, +, \circ}\right)$$.

Ring Product
The distributive operation $$\circ$$ in a ring $$\left({R, +, \circ}\right)$$ is known as the ring product.

Binding Priority
We usually simplify our brackets somewhat, by imposing the rule:

$$a \circ b + c = \left({a \circ b}\right) + c$$

... that is, ring product has a higher precedence than addition.