# Definition:Unit of Algebra

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## Definition

Let $R$ be a commutative ring.

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

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.

## Also see