Definition:Unity (Abstract Algebra)/Ring

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

• Definition:Unity of Semiring
• Definition:Unity of Field

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: $(2)$