Definition:Ring with Unity

Definition
A non-null ring $\left({R, +, \circ}\right)$ is a ring with unity iff the semigroup $\left({R, \circ}\right)$ has an identity element.

Such an identity element is known as a unity.

It follows that such a $\left({R, \circ}\right)$ is a monoid.

Also defined as
Some sources allow the null ring to be classified as a ring with unity.

Also known as
Other names for ring with unity are:
 * Ring with a one
 * Ring with identity
 * Unitary ring
 * Unital ring
 * Unit ring

Some sources simply refer to a ring, taking the presence of the unity for granted.

On, the term ring does not presuppose said presence.