Definition:Unity (Abstract Algebra)

This page is about Unity in the context of Abstract Algebra. For other uses, see Unity.

Definition

Unity of Ring

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).

Unity of Field

Let $\struct {F, , \times}$ be a field.

The identity element of the multiplicative group $\struct {F^*, \times}$ of $F$ is called the multiplicative identity of $F$.

It is often denoted $e_F$ or $1_F$, or, if there is no danger of ambiguity, $e$ or $1$.