Definition:Unity (Abstract Algebra)

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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 of the multiplicative group $\struct {F, \times}$ is referred to as the unity of the field $\struct {F, +, \times}$.

It is (usually) denoted $1_F$, where the subscript denotes the particular field to which $1_F$ belongs (or often $1$ if there is no danger of ambiguity).