Definition:Unity (Abstract Algebra)/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 as it can be confused with a unit, which is a different thing altogether.

Also see

 * Definition:Unity of Semiring
 * Definition:Unity of Field