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.

Such an identity is known as a unity.

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

Alternative definitions
Some sources allow the null ring to be classified as a ring with unity.

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