# 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

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 21$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(2)$ - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$