Definition:Ring with Unity

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Definition

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

Then $\struct {R, +, \circ}$ is a ring with unity if and only if 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 $\mathsf{Pr} \infty \mathsf{fWiki}$, the term ring does not presuppose said presence.


Also see

  • Results about Rings with Unity can be found here.


Sources