Definition:Ring (Abstract Algebra)

Definition
A ring $\struct {R, *, \circ}$ is a semiring in which $\struct {R, *}$ forms an abelian group.

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

Ring Axioms
Note that a ring is still a semiring (in fact, an additive semiring), so all properties of these structures also apply to a ring.

Also defined as
Some sources insist on another criterion which a semiring $\struct {S, *, \circ}$ must satisfy to be classified as a ring:

Such sources then use the term rng (pronounced rung): for a ring without an identity.

However, defines a ring as any structure fulfilling axioms $\text A 0$ - $\text A $, $\text M 0$ - $\text M 1$ and $\text D$, whether or not it has a unity.

The more specific structure which does have a unity is termed a ring with unity.

Other sources define a ring as an algebraic structure $\struct {R, *, \circ}$ which, while fulfilling all the other ring axioms, does not insist on $\text M 1$, associativity of ring product.

Such regimes refer to a ring which does fulfil axioms $\text A 0$ - $\text A 4$, $\text M 0$ - $\text M 1$, $\text D$ as an associative ring.

Also known as
Earlier sources, that is, dating to the early $20$th century, tend to use the term annulus for this concept, but the word ring (at least in this context) is now generally ubiquitous.

Also see

 * Categories of Elements of Ring


 * Definition:Rng


 * A commutative ring is a ring $\struct {R, +, \circ}$ in which the ring product $\circ$ is commutative.


 * If $\struct {R^*, \circ}$ is a monoid, then $\struct {R, +, \circ}$ is called a ring with unity.


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


 * An integral domain is a commutative and unitary ring which has no proper zero divisors.


 * If $\struct {R^*, \circ}$ is a group, then $\struct {R, +, \circ}$ is called a division ring.


 * If $\struct {R^*, \circ}$ is an abelian group, then $\struct {R, +, \circ}$ is called a field.

Generalizations

 * Definition:Semiring (Abstract Algebra)
 * Definition:Ringoid (Abstract Algebra)