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.


Examples

$2 \times 2$ Matrices

Let $S$ denote the set of square matrices of order $2$ whose entries are the set of real numbers.

Then $S$ forms a non-commutative ring with unity whose unity is the matrix $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$.


Also known as

Other names for ring with unity include:

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

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