# Definition:Unity of Ring

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*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 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.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 1$: Introduction: $(1.3)$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.3$: 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: $(2)$