Definition:Unit of Algebra

Definition

Let $\left({A_R, \oplus}\right)$ be a unitary algebra over a ring $R$.

The unit of $\left({A_R, \oplus}\right)$, 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 $\left({A_R, \oplus}\right)$.

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