Definition:Unity of Ring

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Not to be confused with Definition:Unit of Ring.

Definition

Let $\left({R, +, \circ}\right)$ be a ring.

If the semigroup $\left({R, \circ}\right)$ has an identity, this identity is referred to as the unity of the ring $\left({R, +, \circ}\right)$.

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