Definition:Ring (Abstract Algebra)

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

That is, in addition to $\left({R, *}\right)$ 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 $\left({S, *, \circ}\right)$ must satisfy to be classified as a ring:

Such sources refer to what this website calls a ring as a rng (pronounced "rung"): i.e. a "ring" without an "identity".

However, this website specifically defines a ring as one fulfilling axioms $A0 - A4, M0 - M1, D$ only, and instead refers to this more specific structure as a ring with unity.

Other sources define a ring as an algebraic structure $\left({R, *, \circ}\right)$ which, while fulfilling all the other ring axioms, does not insist on $M1$, associativity of ring product.

Such regimes refer to a ring which does fulfil axioms $A0 - A4, M0 - M1, D$ as an associative ring.

Historical Note
According to, the ring axioms were first formulated by  in 1893.

Also see

 * Categories of Elements of Ring


 * Definition:Rng


 * 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 called 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.


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


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


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

Generalizations

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