Definition:Ring (Abstract Algebra)

Definition
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 ring addition, or just addition.

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

The group $$\left({R, +}\right)$$ is consequently known as the additive group of $$R$$.

Some sources write $$\left({R, +}\right)$$ as $$R^+$$ but this can be confused with the set of positive elements $$R_+$$ of an ordered ring.

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.

Element Categories
The elements in a ring are partitioned into three classes:
 * 1) the zero;
 * 2) the units;
 * 3) the proper elements.

Ring Less Zero
It is convenient to have a symbol for $$R - \left\{{0}\right\}$$, that is, the set of all elements of the ring without the zero. Thus we usually use:


 * $$R^* = R - \left\{{0}\right\}$$

Historical Note
According to Ian Stewart, the ring axioms were first formulated by Heinrich Martin Weber in 1893.

Also see

 * A commutative ring is a ring $$\left({R, +, \circ}\right)$$ in which the ring product $$\circ$$is commutative.


 * If $$\left({R^*, \circ}\right)$$ is a monoid, then $$\left({R, +, \circ}\right)$$ is a ring with unity.


 * A commutative and unitary ring is a commutative ring $$\left({R, +, \circ}\right)$$ which at the same time is a ring with unity.


 * If $$\left({R^*, \circ}\right)$$ is a group, then $$\left({R, +, \circ}\right)$$ is a division ring.


 * If $$\left({R^*, \circ}\right)$$ is a abelian group, then $$\left({R, +, \circ}\right)$$ is a field.