Definition:Unit of Algebra

Definition

Let $R$ be a commutative ring.

Let $\struct {A_R, \oplus}$ be a unital 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 usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.

Also known as

The unit of a unital algebra can also be referred to as its multiplicative identity.

Also see

• Definition:Unital Algebra

Sources

• John C. Baez: The Octonions (2002): 1.1 Preliminaries