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.

Additive Group is Abelian
Most treatments of this subject include as an axiom that the additive group of a ring is abelian.

This is true, and can be proved directly from the other axioms.

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:


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

Beware: some sources use $R^*$ to mean the group of units of a ring $R$.

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.