Definition:Unit of Algebra

Definition
Let $R$ be a commutative ring.

Let $\struct {A_R, \oplus}$ be a unitary algebra over $R$.

The unit of $\struct {A_R, \oplus}$, denoted $1_A$, is the identity element of the operation $\oplus$:
 * $\forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$

It is sometimes referred to as the multiplicative identity of $\struct {A_R, \oplus}$.

It is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.

Also see

 * Definition:Unitary Algebra