# 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 $\mathsf{Pr} \infty \mathsf{fWiki}$, the term ring does not presuppose said presence.