# Definition:Unity (Abstract Algebra)/Ring

< Definition:Unity (Abstract Algebra)(Redirected from Definition:Unity of Ring)

Jump to navigation
Jump to search
*This page is about Unity of Ring. For other uses, see Unity.*

*Not to be confused with Definition:Unit of Ring.*

## Definition

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

If the semigroup $\struct {R, \circ}$ has an identity, this identity is referred to as **the unity of the ring $\struct {R, +, \circ}$**.

It is (usually) denoted $1_R$, where the subscript denotes the particular ring to which $1_R$ belongs (or often $1$ if there is no danger of ambiguity).

The ring $R$ itself is then referred to as a ring with unity.

## Also known as

When the **ring** is in fact a **field**, the term **unit** is often used for **unity**.

It is preferred that this is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$ as it can be confused with a **unit**, which is a different thing altogether.

## Also see

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 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 - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(2)$