Definition:Ring with Unity

Definition
Let $\struct {R, +, \circ}$ be a non-null ring.

Then $\struct {R, +, \circ}$ is a ring with unity the multiplicative semigroup $\struct {R, \circ}$ has an identity element.

Such an identity element is known as a unity.

It follows that such a $\struct {R, \circ}$ 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.